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I. CONCEPT OF PROBABILITY<br />
1. From the integers from 2 to 40 inclusive, one integer is selected at random. What is the probability that it is<br />
(a) a perfect square, (b) a prime, (c) a multiple of 9,<br />
(d) a perfect square or a prime, (e) a perfect square or a multiple of 9.<br />
2. Comment on each of the following statements:<br />
(a) "In a tennis championship, there are 32 entrants. The probability that each entrant will win is 1/32"<br />
(b) "If 2 unbiased coins are tossed, since there are 3 possible occurrences - 2 heads, a head and a tail, 2 tails - then the<br />
probability of 2 tails is 1/3".<br />
3. (i) A girl has 5 pairs of shoes, each pair being a different style. If she selects 2 shoes at random, what is the<br />
probability they form a matching pair ?<br />
(ii) Three unbiased coins are tossed, what is the probability of obtaining (a) 3 tails , (b) one tail and two heads.<br />
4. (i) Two dice, each numbered 1 to 6, are tossed. Find the probability that<br />
(a) both show a "4" (b) neither shows a "4"<br />
(c) at least one show a "4" (d) the total score is 9.<br />
(ii) If the dice are thrown n times, find the probability of obtaining<br />
(a) no double "4" (b) at least one double "4" in the n throws.<br />
Hence determine the least value of n for which the probability of at least one double 4 is greater than 1/2.<br />
5. (i) An urn contains 7 red and 4 blue balls. Two balls are drawn without replacement. What is the probability that the<br />
balls are<br />
(a) both red (b) a red and a blue (c) both the same colour.<br />
(ii) Find the corresponding probabilities if the balls are replaced after each draw.<br />
6. (i) By listing all possibilities, find the number of committees of 3 possible from 5 persons A, B, C, D, and E. If all<br />
persons are equally likely to be elected, what is the probability that A is not on the committee.<br />
(ii) At a carnival, the probability that a boy will win the 100 m event is 1/10, and the probability he wins the 200 m<br />
event is 1/8. Find the probability that he wins one of these events.<br />
7. In a group A, there are 7 boys and 3 girls, whilst in another group B, there are 2 boys and 8 girls.<br />
(i) One person is selected at random from each group. Find the probability that (a) both are boys (b) one is a boy and<br />
one is a girl.<br />
(ii) If two persons are selected from either of the original groups A and B, find the probability that (a) both are boys (b)<br />
one is a boy and one is a girl.<br />
8. (i) The probability that a certain variety of plant produces a yellow flower is 1/5. If two plants are selected at<br />
random, what is the probability that:<br />
(a) both produces yellow flowers<br />
(b) only one produces a yellow flower<br />
(c) there are no yellow flower<br />
(d) at least one produces a yellow flower.<br />
(ii) If n of these plants are selected, what is the probability that there is at least one yellow flower. Hence, determine the<br />
least number of plants which need to be chosen so that the probability of at least one yellow flower is greater than<br />
0.99.<br />
9. (i) The probability that a biased coin falls heads is 1/4. Find the probability of at least one head in (a) 2 throws, (b) 3<br />
throws , of the coin.<br />
(ii) Determine the least number of times the coin must be thrown so that the probability of at least one head occurring<br />
is greater than 0.999.<br />
10. One of 5 pens is faulty. They are tested one at a time, what is the probability that the faulty pen is :<br />
(a) 1th first one tested (b) the 2nd one testedt<br />
(b) the rd one tested (d) the last one tested<br />
11. (i) How many possibilities are there if (a) 2 (b) 3 and (c) 4 ordinary dice (numbered 1,2,3,4,5,6) are thrown ?<br />
(ii) Find the probability of total score of 7 with (a) 2 dice (b) 3 dice (c) 4 dice. hint: carefully list these cases giving a<br />
total score of 7 with 2,3,4 dice. Do not write out the sample space).
12. (i) There are two lines of defence against an attacking aircraft, the first is a missile and the second an anti-aircraft<br />
gun. The respective probabilities that the aircraft being hit by the missile and the gun are 2/9 and 1/8, respectively.<br />
What is the probability of hitting the aircraft ?<br />
(ii) Two girls have their birthdays in the same week. It is known that one of the girls was born on a Saturday, write<br />
down the sample space and hence determine the probability that both were born a Saturday.<br />
13. (i) From an ordinary deck of cards, only 12 picture cards are retained. They are shuffled (mixed) and a man draws<br />
2 cards at random. He then announce one of these cards is the King or Spades. What is the probability that he holds<br />
two Kings? (hint: list the possible sample space carefully).<br />
(ii) The 2 cards are replaced and the cards re-shuffled. He then draws 2 cards, and announces that he holds at least<br />
one King. Find the probability that he holds two Kings in his hand.<br />
14. An a small raffle, there are only 2 prizes, and 30 tickets are sold. A boy has 3 tickets in the raffle. Find the<br />
probability that he wins (a) first prize, (b) both prize, (c) only the 2nd prize, and (d) a prize.<br />
15. In a raffle in which 200 tickets are sold, there are 3 prizes - first prize of $100, second prize of $50, third prize of<br />
$30.<br />
(i) A girl has one ticket in the raffle. What is the probability that she wins (a) a prize, (b) at least $50.<br />
(ii) A man has 2 tickets in this raffle. WHat is the probability that he wins (a) a prize, (b) both prizes.<br />
16. In a bag of balloons, the ratio of red balloons to other colours is 1:5, if 3 balloons are selected from the bag at<br />
random, what is the probability that (a) at least one balloon is red, (b) exactly one balloon is red.<br />
17. If the probabilities that three students A, B and C may pass the HSC are 4/7, 3/5, and 2/3, respectively. What is the<br />
probability that<br />
(a) all 3 pass (b) all 3 fail<br />
(c) two students only pass (d) only one student passes<br />
(e) at least two pass (f) at least one passes.<br />
18*. A point is moving randomly on a line of length L. What is the probability that the ratio of the shorter to the longer<br />
segment is less than 1/3 ?<br />
19*. A point is moving randomly inside an equilateral triangle whose side length is 3 cm. What is the probability that<br />
its distance to any corner is greater than 1 ?<br />
20*. Two persons agree to meet at between 2 PM to 4 PM, but each of them will wait 30 minutes for the late comer.<br />
What is the probability that they will meet ?<br />
ANSWERS: 1.(a) 5/39 (b) 4/13 (c) 4/39 (d) 17/39 (e) 3/13.<br />
3.(i) 1/9 (ii) (a) 1/8 (b) 3/8 4.(i)(a) 1/36 (b) 25/36 (c) 11/36 (d) 1/9 (ii) (a) (35/36)n<br />
(b) 1-(35/36)n; min n is 25 5. (i) (a) 21/55 (b) 28/55 (c) 27/55 (ii) (a) 41/121 (b) 56/121<br />
(c) 65/121. 6. (i) ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE; 4/10=2/5<br />
(ii) (7+9)/(80=1/5. 7.(i) (a) 7/50 (b) 31/50 (ii) (a) 11/45 (b) 37/90.<br />
8.(i) (a) 1/25 (b) 8/25 (c) 16/25 (d) 9/25 (ii) 1-(4/5)n least n is 21.<br />
9.(i) (a) 1-3/4)2 = 7/16 (b) 1-(3/4)3 =37/64 (c) 1-(3/4)n >0.999 when (4/3)n >1000; least<br />
n is 25. 10.(a) 1/5 (b) 1/5 (c) 1/5 (d) 1/5. 11. (a) 36 (b) 216 (c) 1296<br />
(ii) (a) 1/6 (b) 5/72 (c) 5/324. 12.(i) 23/72 (ii) 1/13. 13.(i) 3/11 (ii) 3/19. 14.(a) 1/10<br />
(b) 1/145 (c) 27/290 (d) 28/145. 15. (i) (a) 3/200 (b) 1/100 (ii) (a) 297/9950 (b) 1/19900<br />
16.(a) 91/216 (b) 25/72. 17.(a) 8/35 (b) 2/35 (c) 46/105 (d) 29/105 (e) 2/5 (f) 33/35.<br />
18.1/2 19. 20. 7/16.<br />
PERMUTATION I (LINEAR ARRANGEMENTS)<br />
1. How many different arrangements can be made by taking 3 letters of the word SUNDAY ?<br />
2. In how many ways can 5 boys be arranged in a row ?<br />
3. In how many ways can a first, second and third prize be awarded in a class of 8 students ?<br />
4. How many 5-digit numbers can be formed from the digits 2,3,5,6,8,9 if no digit can be used more than once in a<br />
number ? How many even numbers can be formed ?<br />
5. Find the value of (i) (ii) (iii)
6. Find the value of (i) 6! (ii) 8! / 5!<br />
7. In how many ways can 4 consonants and 3 vowels be arranged in a row (a) so that the 3 vowels are always together,<br />
(b) so that the first and the last places are occupied by consonants.<br />
7a. There are 4 different mathematics books, 6 different physics books, and 2 different chemistry books are to be<br />
arranged on a shelf. How many different arrangements are possible if (a) the books in each category must all stand<br />
together , (b) only the mathematics books must stand together.<br />
8. In how many ways can four girls and three boys be arranged in a row so that (a) the boys are always together, (b)<br />
the girls and boys occupy alternate places ?<br />
9. In how many ways can 6 people be seated in a motor car if only 2 can occupy the driver's position ?<br />
10. In how many ways can 6 people be arranged in a circle if 2 particular people are always (a) together (b)<br />
separated.<br />
11. Father, mother and 6 children stand in a ring. In how many ways can they be arranged if father and mother are not<br />
to stand together.<br />
12. In how many ways can 5 boys and 3 women be arranged (a) in a row , (b) in a circle if both cases the women are<br />
always to stand together ?<br />
13. Four men and four women are to be seated alternately (a) in a row (b) at a round table. In how many ways can this<br />
be done ?<br />
14. How many even number of 4 digits can be forned with the figures 3,4,7,8 if (a) no figure is repeated , (b) repetitions<br />
are allowed.<br />
15. How many numbers greater than 4000 can be formed from the figures 3,5,7,8,9 ? (repetitions not allowed).<br />
16. In how many ways can the letters of the word PERMUTE be arranged if (i) the first and last places are occupied by<br />
consonants, (ii) the vowels and consonants occupy alternate places.<br />
17. If , find r.<br />
18. The number of arrangements of 2n+2 different objects taken n at a time is to the number of arrangements of 2n<br />
different objects taken n at a time as 14:15. Find the value of n.<br />
19. How many numbers of 7 digits can be formed from the digits 1,2,3,4,5,6,7 if (i) the numbers begins with 1 and ends<br />
with 2, (ii) there are not more than 2 digits between 1 and 2 ?<br />
20. If , find the value of n. [4]<br />
21. In how many ways can 5 different mathematics books, 4 different physics books and 2 different chemistry books be<br />
arranged on a shelf if the books in each subject are to be together.<br />
22. In how many can 3 men, 3 women and 3 boys be arranged in a row if the three boys are to remain together ?<br />
23. Find the number of arrangements of the letter in the word PENCILS if (i) E is next to I, (ii) E precedes I, (iii) there<br />
are three letters between E and I.<br />
24. In how many ways can the letters of the word PRINCIPLE be arranged? In what proportion of these arrangements<br />
do the letters P come together ? .<br />
25. In how many ways can the letters in PRECISION be arranged? In how many of these arrangements do the vowels<br />
occupy even places ?<br />
26. How many arrangements can be made by the letters of word DEFINITION (i) if the letters I do not occupy the first<br />
or last place, (ii) if the letters I are together?<br />
27. How many different arrangements of the letters in TOMATO are there, if the letters O are to be separated?<br />
28. A car can hold 3 people in the front seat and 4 in the back seat. In how many ways can 7 people be seated in the<br />
car if 2 particular people must sit in the back seat and 1 particular person is the driver?<br />
29. In how many ways can 4 people be accommodated if there are 4 rooms available?<br />
30. In how many ways can 8 oarsmen be seated in a eight-oared boat if 3 can row only on the stroke side and 3 can<br />
row only on the bow side ?<br />
31. Prove that (i) from the definition of .<br />
(ii) the formula for that .<br />
32. Prove that .<br />
33. In how many ways can 5 men and 5 women by arranged in a circle so that the men are separated? In how many<br />
ways can this be done if two particular women must not be next to a particular men.<br />
34. How many arrangements of the letters of the word PARRAMATTA are possible?<br />
ANSWERS:<br />
1. 120 2. 120 3. 336 4. 720; 360 5.(i) 120 (ii) 11800 (iii) 6720 6.(i) 720 (ii) 336<br />
7.(a) 720 (b) 1440 7a (a) 207360 (b) 8709120 8.(a) 720 (b) 144 9. 240 10.(a) 120<br />
(b) 48 (c) 72 11. 3600 12. (a) 4320 (b) 720 13.(a) 1152 (b) 144 14. (a) 12 (b) 128<br />
15. 216 16 (i)1720 (ii) 72 17. r=3 18. 3 19.(i) 120 (ii) 15 20. 4 21. 34560 22. 30240
23. (i) 1440 (ii) 2520 (iii) 120 24. 90720, 22.2% 25. 181440; 1440 26. (i) 141120 (ii) 20160 27. 120 28. 288 29. 256<br />
30. 1152 33. 2880; 864 34. 37800<br />
PERMUTATION II: ARRANGEMENTS IN A CIRCLE<br />
1. In how many different ways can<br />
(i) 4 persons be arranged (a) in a line (b) around a circle ;<br />
(ii) 7 persons be arranged (a) in a row (b) in a ring;<br />
(iii) 8 different table mats must be placed (a) along one side of a rectangular table (b) around an oval table.<br />
[Ans: (i) (a) 4!=24 (b) 3!=6 (ii) (a) 7!=5040 (b) 6!=720 (iii) (a) 8!=40320 (b) 5040]<br />
2. In how many ways can 4 boys and 4 girls be arranged in a ring if<br />
(i) there are no restriction on where the boys and girls are placed;<br />
(ii) the boys and girls are alternate;<br />
(iii) a particular boy P is next to a particular girl Q;<br />
(iv) two particular girls R,S do not wish to be placed next to each other;<br />
(v) the particular boy P is to be placed between the particular girls R, S;<br />
(vi) three friends A,B,C wish to be together;<br />
(vii) the boys and girls are to be in separate groups.<br />
[ans: (i) 5040 (ii) 3!4!=144 (iii) 2(6!)=1440 (iv) 5(6!)=3600 (v) 2(5!)=240 (vi) 3!5!=720 (vii) 4!4!=576]<br />
3. Eight men and two women are to be seated around a table. In how many ways can they be arranged if<br />
(i) each can choose any seat;<br />
(ii) the eldest person is to have the most comfortable chair;<br />
(iii) the two women are to seat together;<br />
(iv) the two women are not sit directly opposite one another ;<br />
(v) a partcular man is not to sit between the women.<br />
[ans: (i) 9!=362880 (ii) 362880 (iii) 2(8!)=80640 (iv) 8(8!)=322560 (v) 9!-2(7!)=352880]<br />
4. The numbers 1,2,3,4,5,6 are to be arranged around a circle.<br />
(a) How many ways can this be done ? How many of these arrangements have<br />
(i) the odd and even numbers altering;<br />
(ii) at least two odd numbers together;<br />
(iii) all the odd numbers together;<br />
(iv) the numbers 5,6 together but the number 4 is not next to either 5 or 6 ?<br />
(b) What is the probability that if one of these arrangements of the numbers 1,2,3,4,5,6 in a circle is selected random,<br />
then this arrangement has<br />
(i) the odd and even numbers altering;<br />
(ii) at least two odd numbers occurring together;<br />
(iii) all the odd numbers together;<br />
(iv) the numbers 5,6 together but neither next to the number 4?<br />
[ans: (a) 5!=120 (i) 2!3!=12 (ii) 120-12=108 (iii) 3!3!=36 (iv) 2.2.(3!)=24 (b) (i) 1/10 (ii) 9/10 (iii) 3/10 (iv) 1/5]<br />
5. Ten marbles (indistinguishable except for colour) are to be arranged in a circle on the ground. In how many different<br />
ways can this be effected when<br />
(i) the marbles are all of different colours;<br />
(ii) four of the marbles are of the same colour and the rest are different;<br />
(iii) there are 6 red, 3 yellow, 1 white marbles.<br />
[ans: (i) 362880 (ii) 9!/4!=15120 (iii) 9!/(6!3!)=84]<br />
6. In how many ways can<br />
(i) 8 different coloured beads be arranged on a string to form a necklace;<br />
(ii) 7 unlike keys be placed on a key ring<br />
(iii) 4 green, 1 pink, 2 blue beads form a circular bracelet.<br />
[ans: (i) 7!/2=2520 (ii) 6!/2=360 (iii) 6!/(4!2!)=15; beads indentical except colour]<br />
7. 12 counters (each containing a hole through the centre, and identical except for colour) are to be placed<br />
(a) in a line (b) about a circle (c) on a bracelet<br />
In how many ways can this be done if<br />
(i) the counters are all different in colours;<br />
(ii) there are 7 purple, 4 black, 1 green counters.
[ans: (i)(a) 12! (b) 11! (c) 11!/2 (ii)(a) 12!/(7!4!) (b) 111!/(7!4!) (c) 11!/(7!4!2!)]<br />
8. It is required to seat n people at a round table. How many different possible arrangements are there ?<br />
How many arrangements are possible if (i) three specified people are to sit together ?<br />
Use these results to show that if n people are given seats at random at a round table, the chance of two particular<br />
people aitting together is 2/(n-1) and the chance of a group of three particular people sitting together is 6/(n-1)(n-2).<br />
[ans: (n-1)! (i) 2[(n-2)!] (iii) 3!(n-3)!]<br />
9. (i) A dinner party consisting of 5 couples, sit around a rectangular table, with ladies and gentlemen altering. The<br />
host and hostess each occupy one end of the table and their guests are arranged four on each side. Find the number of<br />
ways the party can be seated. [ans 4!4!=576]<br />
(ii) At a meeting, there are 6 sporting associations represented. Each association is allowed 1 delegate to attend the<br />
meeting, as well as 2 assistants. In how many ways can the persons attending the meeting be seated at a large round<br />
table if the assistants must sit on either side of the delegate in each case. [ans 2^6.5!=7680]<br />
10. 9 persons are arranged in a straightline. What is the probability that two particular persons are side by side ? What<br />
is the corresponding probability if the nine persons are arranged in a circle (instead of in a line)? [ans 2(8!)/9!=2/9;<br />
2(7!)/8!=1/4]<br />
11. In how many ways can ten girls be placed around a table if two particular girls must be separated ? The ten girls<br />
take their places at random about the table. What is the probability these 2 particular girls are NOT together ? [ans<br />
7(8!)=282240; 7(8!)/9=7/9]<br />
12. In how many ways can 6 men and 6 women be seated about a round table ? If these men and women take their<br />
seats at random, determine the probability that<br />
(a) the men and women sit alternately at the table;<br />
(b) a particular couple are seated together;<br />
(c) two particular women are not sitting side by side;<br />
(d) a particular woman is not seated next to either of 2 particular men;<br />
(e) two particular men are seated on either side of a particular woman;<br />
(f) three particular persons are seated together.<br />
[ans: 11! (a) 6!5!/11!=1/462 (b) 2(10!)/11!=2/11 (c) 1-2/11=9/11 (d) 9.8(9!)/11!=36/35 (e) 2(9!)/11!=1/55 (f)<br />
3!9!/11!=3/55]<br />
13. Nine beads (indistinguishable except for colour) are to be arranged<br />
(a) in a line (b) around a circle (c) on a necklace<br />
In how many ways can this be done if<br />
(i) the beads are all of different colours;<br />
(ii) there are 4 red, 3 green, 1 yellow and 1 blue bead.<br />
[ans: (i) (a) 362880 (b) 40320 (c) 8!/2=20160 (ii) (a) 9!/4!3!=2520 (b) 8!/4!3!=280 (c) 140.]<br />
14. The numbers 0,1,2,3,4,5,6,7,8,9 are to be arranged in a circle in all possible orders. If one of these arrangements is<br />
selected at random, what is the chance that<br />
(i) the numbers 8,9 are together;<br />
(ii) the number 8,9 are separated;<br />
(iii) the numbers 4,5,6 are together;<br />
(iv) it is the arrangement 0123456789 ;<br />
(v) all the even numbers are together;<br />
(vi) the even and odd numbers alternate ;<br />
(vii) at least two even numbers are together.<br />
[ans: (i) 2/9 (ii) 1-2/9=7/9 (iii) 3!7!/9!=1/12 (iv) 1/9!=1/362880 (v)5!5!/9!=5/126 (vi) 4!5!/9!=1/126 (vii) 1-<br />
1/126=125/126]<br />
15. (i) There are 5 boys and 5 girls in a gymnastics squad. In how many ways can they be formed into<br />
(a) one ring, with no girls next to one another;<br />
(b) two concentric rings, with a girl behind each boy ?<br />
(ii) Show that 10 persons may be seated at two round tables, 5 persons at each, in 10!/5^2 ways.<br />
[ans: (a) 4!5!=2880 (b) 4!5!=2880 (ii) answer is ]<br />
IV. COMBINATION
1. In how many ways can 3 books be selected from 8 different books? [56]<br />
2. In how many ways can a set of 2 boys and 3 girls be selected from 5 boys and 4 girls? [40]<br />
3. How many different selections can be made by taking 3 of the digits 4, 5, 6, 7, 8, 9 ? [20]<br />
4. In how many ways can a committee of 4 men and 5 boys be formed from 8 men and 7 boys ? [1470]<br />
5. From 8 soldiers, 7 sailors and 5 airmen how many sets can be formed each containing 5 soldiers, 4 sailors and 3<br />
airmen ? [19600]<br />
6. In how many ways can 8 boys be divided into two sets containing 5 and 3 respectively? [56]<br />
7. A committe of 6 is to be selected from 10 people of whom A and B are two. How many committees can be formed (i)<br />
containing A and B, (ii) excluding A if B is included ? [70;140]<br />
8. In how many ways can 3 cards be selected from a pack of 52 playing cards if [i] at least one of them is an ace, [ii]<br />
not more then one is an ace ? [4804; 21808]<br />
9. In how many ways can a commitees of three men and 4 women be chosen from 6 men and 7 women? What<br />
proportion of these commitees contains a particular man and a particular women? [700; 1/7]<br />
10. A commitees of 7 politicians is chosen from 10 liberal members, 8 labor members and 5 independents. In how many<br />
ways can this be done so as to include exactly 1 independent and at least 3 liberal members and at least 1 labor<br />
member? [73080]<br />
11. An eleven is to be chosen from 15 cricketers of whom 5 are bowlers only.2 others are wicket keepers only and the<br />
rest are batsmen only.How many possibles elevens can be chosen which contain ( i ) 4 bowlers, 1 wicketkeeper and 6<br />
batsmen, ( ii )at least 4 bowlers and at least 1 wicktkeeper? [280; 742]<br />
12. From 7 teachers and 5 pupils a commitee of 7 is to be formed. How many commitees can be selected if both<br />
teachers and pupils are represented and the teachers are in a majority? [595]<br />
13. From 4 oranges, 3bananas and 2 apples, how many selections of 5 pieces of fruit can be made, talking at least one<br />
of each kind? [98]<br />
14. In lotto, a frame contain the numbers 1 to 40 and we can select any 6 of these numbers. How many such<br />
combinations are possible? [3,838,380]<br />
15. In TAB betting, the "trifecta" pays on first three horses in correct order, the "quinella" pays on the first two horses<br />
in either orther. In a 10 horses race what is the possible number of (a) trifecta combination (b) quinella combinations?<br />
[720;45]<br />
16. In how many ways can a jury of 12 be chosen from 10 men and 7 women so that there are at least 6 men and not<br />
more than 4 women on each jury? [1946]<br />
17. In how many ways can a set of 3 or more be selected from 9 people? [466]<br />
18. In how many ways can n things be shared between 2 people. [2n - 2]<br />
19. In how many ways can a commitee of 3 men and 4 boys be chosen from 7 men and 6 boys so as not to include the<br />
youngest boy if the eldest man is serving? [375]<br />
20. How many ( i ) selections, ( ii ) arrangements consisting of 3 consonants and 2 vowels can be made from 8<br />
consonants and 4 vowels? [336; 40320]<br />
21. In how many ways can 4 Physics books and 3 Mathematics books be arranged on a shelf if a selection is made from<br />
6 different Physics books and 5 different Mathematics books? In how many of these arrangements are the Physics<br />
books together? [756,000; 86400]<br />
22. In how many ways can 3 boys and 2 girls be arranged in a row if a selection is made from 5 boys and 4 girls? In<br />
how many of these arrangements does a boy ocupy the middle position? [7200; 4320]<br />
23. How many words (arrangement of letters ), containing 3 consonants and 2 vowels, can be formed from the letters of<br />
the word promise ? [1440]<br />
24. From the definition of nCr prove each of the following:<br />
(a)<br />
(b)<br />
(c)<br />
(d)<br />
(e)<br />
25. If , find the value of n. [10]<br />
26. The ratio of the number of combinations of (2n+2) different objects taken n at a time to the number of combinations<br />
of (2n-2) different objects taken n at a time is 99:7. Find the value of n. [5]<br />
27. In how many ways can 9 books be distributed amongst a man, a woman and the child, if the man receives 4, the<br />
woman 3 and the child 2? [1260]<br />
28. In how many ways can 8 boys be divided into two unequal sets? [92]<br />
29. In how many ways can 8 girls be divided into 4 sets of 2? [105]
V. PROBABILITY PROBLEMS ASSOCIATED WITH PERMUTATION AND COMBINATION<br />
1. Five cards are drawn at random from a pack of 52 playing cards . what is the probability that they are all from the<br />
same suit? [33/16660]<br />
2. A bag contain 5 red balls and 4 wite balls. Three balls are withdrawn without replacement. Find the probability of<br />
drawing at least 2 red balls [25/42]<br />
3. An urn contains three white balls, 4 red balls and 5 black balls. Three balls are drawn at random. what is the<br />
probability that they are<br />
(a) different colour (b) the same colour? [3/11; 3/44]<br />
4. From7 teachers and 5 pupils, a random selection of 7 is make. what is the probability that is contains at least 4<br />
teachers? [149/198]<br />
5. A committee of 3 men and 4 women is to be chosen from 6 men and 7 women. what is the probability that is contains<br />
a particular manand a particular woman? [2/7]<br />
6. Five cards are drawn from a pack of 52 playing carts. What is the probability of drawing at leas 3 aces? [19/10829]<br />
7. Eight people are to be divided into two groups. What is the probability that there will be 4 in each groups? [35/81]<br />
8. The letters of the word "PROMISE" are arranged in arow. What is the probability that there are 3 letters between P<br />
and R ? [1/7]<br />
9. Six people arrange themselves at random in a circle. what is the probability that the tallest and the shortest are<br />
toghether? [2/5]<br />
10. Four men and three boys are arranged in a straight line. What is the probability that the men and the boys occupy<br />
alternate position? [1/35]<br />
11. The number plates of the motor car contain 3 letters of the alphabet followed by 3 numerals. How many such<br />
number plates can be made? What proportion of these would contain 3 letters the same and 3 numerals the same ?<br />
[17576000;1/67600]<br />
12. The letters of the word "independence" are arranged in a row. What is the probability of the letters "e" being<br />
toghether? [1/55]<br />
13. A commitee of 6 is to be selected from 10 people. What is the probability of the youngest and oldest being on the<br />
same commitee? [1/3]<br />
14. A party of twelve, of whom A and B are two, are arranged at random in a straight line. What is the probability that<br />
A and B are not next to one another? [5/6]<br />
15. An urn contains 5 red cubes and 4 white cubes. Three cubes are drawn in succession without replacement. What is<br />
the probability that<br />
(a) the fisrt two cubes are red and the third one white.[10/63]<br />
(b) any two cubes are red, and one is white? [10/21]<br />
16. A carton contains 15 transistors of which 5 are defective. If a random sample of 6 transistors is drawn from the<br />
carton (without replacement) determine the probability of 0, 1, 2, 3, 4, 5 red cubes in the sample<br />
[0.0419;0.2517;0.4195;0.24;0.045;0.002]<br />
17. An urn contains 12 distinguishable cubes of which 5 are red and remainder black. If a random sample of 6 cubes is<br />
drawn without replacement, calculate the probabilities of 0,1,2,3,4,5 red cubes in the sample.<br />
[1/132;5/44;25/66;25/66;5/44;1/132]<br />
18. Two boxes each contain eight balls. In box A there are 3 black and 5 white balls, in box B there are 1 black and 7<br />
white balls. For each box find the probability that two balls chosen at random without replacement will both be white.<br />
[5/14;3/4]<br />
19. A sample of 3 coins is selected without replacement from 8 coins, consisting of 4 five cents coins and 4 ten cents<br />
coins. What is the probability the sample contains at least 2 five-cent coins? [1/2]<br />
20. A hand of 5 cards is dealt from a pack of 52 playing cards. What is the probability that it will contain at least one<br />
ace? [0.3412]<br />
21. From a group of 12 people of whom 8 are males and 4 are females, a sample of 4 is selected at random. what is the<br />
probability that the sample contains at least 2 females? [67/165]<br />
22. Urn A contain 6 white and 4 black balls. Urn B contains 2 white and 2 black balls.From urn A two balls are<br />
selected at random and placed in urn B. From urn B two balls are then selected at random.What is the probability that<br />
exactly one of these two balls is white? [128/225]<br />
23. From a set of 10 cards numbered 1 to 10, two cards are drawn without replacement. What is the probability that (a)<br />
both numbers are even, (b) one is even and the other is odd, (c) the sum of the two numbers is 12, (d) both numbers are<br />
even and the sum of two numbers is 12? [2/9; 5/9; 4/45; 2/45]<br />
24. The letters of the word TOMATO are arranged at random in a row.What is the probability that (a) two letters O are<br />
together, (b) the two letters O are not together? [1/3; 2/3].
25. The letters of the word TUESDAY are arranged at random in the row.What is the probability that (a) the vowels and<br />
consonants occupy alternate positions (b) the vowels are together (c) the vowels are together and the letter T occupies<br />
the first place? [1/35; 1/7; 1/35]<br />
26. Four girls and four boys arrange themselves at random in (a) a row, (b) a circle. What is the probability in each<br />
case that the girls and the boys occupy alternate position?<br />
27. Six people, of whom A and B are two, arrange themselves at random in a row.What is the probability that<br />
(a) A and B occupy the end position, [1/15]<br />
(b) A and B are not next to each other, [2/3]<br />
(c) there are at least three people between A and B? [1/2]<br />
28. The digits 1, 2, 3, 3, 4, 5, 6 are used to form numbers which contain two or more digits.( The same digit can not be<br />
used more than once in a number ). What proportion of such numbers are even numbers? [1/2]<br />
29. Five cards are selected without replacement from a pack of 52 playing cards. What is the probability of (a) exactly<br />
3 hearts, (b) 4 aces, (c) no hearts? [2717/33320;1/54145;2109/9520]<br />
VI. BIONOMIAL DISTRIBUTION - PART I<br />
1. Write out the expansion of .<br />
(i) the probability p that A biassed coin fall heads is 2/3; what is the probability q that it does not fall heads?<br />
(ii) this coin is tossed 3 times. Find the probability that there will be :<br />
(a) 0 head (b) exactly 1 head<br />
(c) exactly 2 heads (d) 3 heads<br />
What is the sum of these probabilities? comment on this.<br />
2. A true- false quiz consists of 5 quesions. A student selects answers at random. What is the probability of his being<br />
correct in:<br />
(a) all 5 questions (b) exactly 2 questions<br />
(c) at most 2 questions (d) at least e questions?<br />
3. If the births of boys and girls are assumed to be equally likely, what is the probability that in a family of 8 children,<br />
there is<br />
(a) exactly 4 girls (b) all girls<br />
(c) at least 2 girls (d) a majority of girls?<br />
4. A bag contians 8 white balls and 12 black balls. Balls are withdrawn one at a time, the ball being replaced after<br />
each draw.<br />
There are 4 drawings. What is the probability of drawing out:<br />
(a) all black balls (b) at least 3 white balls?<br />
5. An ordinary die is rolled 3 times. If X denotes the appearance of a "4" on the upper face of the die, find the following<br />
probabilities<br />
(a) p(x=2) (b) p(x>2) (c) p(x=0)<br />
6. At a referendum 662/3% of voters were in favour of a certain proposal.A television interviewer approached 7 voters<br />
chosen at random. Find the probability that the number in favour of the proposal was:<br />
(a) exactly 3 (b) no more than 1.<br />
7. (i) A card is selected at random from a regular pack the card is replaced after the selection. What is the probability<br />
that it is a diamond.<br />
(ii) If 5 cards are drawn at random one at a time with replacement, from the pack, find the probability of drawing:<br />
(a) exactly one diamond (b) a majority of diamonds<br />
8. An urn contains 4 blue, 3 green and 2 yellow marbles. Marbles are drawn out, on at a time, with replacement. If 6<br />
drawings are made , find the probabilioty of obtaining<br />
(i) all green marbles (ii) exactly 4 green marbles<br />
9. For a biassed coin, the probability of a head is given by p= 0.3. If the coin is tossed 4 times, find the probability of<br />
(a) 3 heads exactly (b) at most 3 heads.<br />
10. It is known that at noon the sun is hidden by clouds on an average of two days out of every three. If 5 consecutive<br />
days are taken, find the probability of the sun shining at noonon:<br />
(a) each day (b) the first 4 days only<br />
(c) 4 of the days (d) at least 4 days.
11. In a bag, there are three counters marked with the digit "3" and five counters marked with digit "2". Four counters<br />
are drawn out of the bag one at a time with replacement after each draw, find the probability that:<br />
(a) all counters drawn are marked with the digit "2"<br />
(b) two of the counters are marked with "2" and two with "3"<br />
(c) the sum of the digits on the counters is greater than 10.<br />
12. For a certain die, the probability of a "6" times in succession. Find the probability that a "6" appears:<br />
(a) on no occasion (b) exactly 5 times<br />
(c) no less than 5 times (c) no more than 5 times<br />
13. Two ordinary dice are tossed simultaneously on the table. What is the probability p of a total of 7? If 4 players each<br />
roll a pair of dice, determine the probability that<br />
(a) only one roll a 7 (b) at least 2 roll a 7<br />
(c) none of them rollr a 7 (d) all of them roll a 7<br />
14. Suppose we take a random sample of 5 items from a large number of items which are known to be 10% defective.<br />
What is the probability that our sample contains<br />
(a) all defectives (b) no defectives?<br />
Show that the chance our sample contains more than 2 defectives is less than 1%<br />
15. (i) A machine produces parts which have a 90% chane of meeting the required speccification. A sample of 10 parts<br />
is taken, find the probability that exactly 2 of these fail to meet the spectifications required.(Give answer correct to 4<br />
significant figures)<br />
(ii) An event has a 10% chance of occurring. Evaluate to 3 decimal places, the probability of it occurring at least once<br />
in 10 independt trials.<br />
ANSWERS: 1. (i) 1/3 (ii) (a) 1/27 (b) 2/9 (c) 4/9 (d) 8/27; 1, one of these events is certainly to<br />
occur hence the probability is 1. 2. (a) 1/32 (b) 5/16 (c) (1+5+10)/32 (d) 13/16 3.(a) 70/256 (b) 1/256 (c) 247/256 (d)<br />
(56+28+8+1)/256 4.(a) 81/625 (b) (16+96)/625 5.(a) 15/216 (b) (15+1)/216 (c) 125/216 6.(a) 280/2187 (b)<br />
(1+14)/2187 7.(i) 1/4 (ii) (a) 405/1024 (b) (90+15+1)/1024 8.(i) 1/729 (ii) 60/729 9.(a) 4(0.7)(0.3)3=0.0756 (b) 1-<br />
(0.3)4 =0.9919 10.(a) 1/243 (b) 2/243 = (1/3)4(2/3) ; not binomial (c) 10/243 (d) (10+1)/243 11.(a) 625/4096 (b)<br />
1350/4096 (c) (540+81)/4096 12.(a) 1/4096 (b) 1458/4096 (c) (1458+729)/4096 (d) 1-(3/4)6 = 3367/4096 13. 6/36 (a)<br />
500/1296 (b) 1- (625+500)/1296 (c) 625/1296 (d) 1/1296 14.(a) 0.00001 (b) 0.59049 ; prob.=0.00856 < 0.01 i.e 1%<br />
15.(i) 0.1937 (ii) 1-(0.9)10 = 0.651<br />
BINOMIAL PROBABILITY<br />
1. In 12 throws of a die, what is the probability of exactly 4 sixed turning up?<br />
2. Hospital records show that of patients suffering from a certain complaint, 75% die of it. What is the probability that,<br />
of 6 randomly selected patients, all will recover?<br />
3. A marksman finds that, on the average, he hits the target 4 times out of 5. If he fires 4 shots, what is the probability<br />
of (i) more than 2 hits, (ii) at least 3 misses.<br />
4. What proportion of families with exactly 6 children should be expected to have at least 3 boys? (Assume that a child<br />
is a boy is 1/2 and ignore the possibility of multiple births.)<br />
5. A manufacturer of metal pistons finds that on the average 12% of his pistons are rejected because they are either<br />
oversized or undersized, What is the probability that a batch of 10 pistons will contain (i) no more than 2 rejects, (ii) at<br />
least 2 rejects.<br />
6. It is known that, in the long run, 7 out of 10 students from year 12 of a class will enter university. Assuming a<br />
binomial distribution, find the probability that out of a group of 5 students chosen at random from this form (i) exactly<br />
3 enter the university, (ii) any number up to and including 3 will enter, (iii) only the first 3 chosen will enter.<br />
7. In the long run 40% of patients treated for particluar disease with the drug streptomycin are cured. If a doctor were<br />
to treat 10 randomly chosen patients with the drug, what is the probability of (i) 9 cures, (ii) more than 8 cures ?<br />
A second doctor treats 10 patients with a new drug and obtains more than 8 cures. Comment on this result.
8. A dental inspector find that about 20% of children of a certain area have tooth decay. What is the probability that<br />
out of 6 children examined, only the first, third and fifth will have tooth decay.<br />
9. A long series of hospital record show that 40% of cases of a certain disease fail to recover. Calculate the probability<br />
of a sample of 10 patients yielding only one death. If it were known that these ten patients had been subjected to a new<br />
treatment, what inference would you make about the efficacy of this treatment.<br />
10. A Gallup Poll established that 60% of people interviewed are in favour of a certain proposal. What is the<br />
probability that if 15 people are selected at random, there will be a majority of them against the proposal.<br />
11. Three uniform tetrahedra (triangular pyramids) each has one face black, another white, another red and the other<br />
green, When tossed onto a table, 3 faces of each tetrahedra can be seen. What is the probability that (i) no black face<br />
can be seen, (ii) exactly 2 black faces can be seen, (iii) at least 2 red faces can be seen, (iv) 3 white faces and only 1<br />
green face can be seen.<br />
12. A die is rolled 10 times. What is the probability of at least 2 sixes turning up?<br />
13. A die is loaded such that in 10 independent trials the probability that an even number appears 5 times is twice the<br />
probability that an even number appears 4 times. What is the probability that an even will appear in a single trial?<br />
14. What is the probability that a family of 4 children will contain 4 girls.<br />
15. The probability that persons aged 60 and 65 will survive the next year is 0.97 and 0.96 respectively. From a group<br />
of 5 people of whom 3 are aged 60 and 2 aged 65, what is the probability that (i) not more than one will die within next<br />
year, (ii) the two aged 65 survive and not more than one dies within the next year.<br />
16. How many times must a fair coin be tossed so that the probability of at least one head exceed 0.9?<br />
17. In an examination consisting of 10 true-false questions, a student guesses the answers. What is the probability that<br />
he will gain 70% or better?<br />
18. Assuming that the chance of death by accident is 1/300, estimate the probability that out of random selection of<br />
150, there will be at least 1 death by accident.<br />
19. John estimates that the probability of his winning any one game of tennis against a particular opponent is 1/3. How<br />
many games should they play so that the probability that John wins at least one gam is greater than 0.9?<br />
20. On the average, a typist has to correct one word out of 800 words. Assuming that a page contains 200 words, find<br />
the probability of more than one correction per page.<br />
21. Steel rods are manufactured to be 4 cm in diameter but they are acceptable if they are between 3.98 cm and 4.02<br />
cm. The manufacturer find that in the long run 3% are rejected as oversized and 3 % are undersized.<br />
A random sample of 10 rods is selected randomly from the population. What is the probability that it contains not more<br />
than one which has to be rejected.<br />
22. Four families each have 4 children. What is the probability that (i) at least one of these families has 2 boys and 2<br />
girls, (ii) each family has at least 1 boy.<br />
23. A man finds that on the average he hits the target 9 times out of every 10 times and scores a bull's eye on the<br />
average once every 5 rounds. He fires 4 rounds. What is the probability that (i) he hits the target each time, (ii) he<br />
scores at least 2 bull's eyes, (iii) he scores at least 2 bull's eyes and he has hit the target on each of the 4 rounds.<br />
24. In a packet of flower seeds 40% are known to be pink flowering and the remainder yellow. Calculate the probability<br />
of getting 0, 1, 2, ...,6 pink flowers in a row of 6 plants. If 2500 rows each of 6 plants are planted, how many rows<br />
would be expected to contain (a) all pink flower, (b) all yellow flowers.<br />
25. If 10 coins are tossed 50 times, in how many cases would we expect the number of heads to exceed the number of<br />
tails.<br />
26. On the average, a man scores a bull's eyes in 2 out of every 3 shots. He fires at a target in multiples of 5 shots.<br />
Estimate how many mulriples of 5 shots with 3 bull's eyes would occur in a thousand multiples.
27. A factory has 7 machines, 4 of model A which are in use, on the average, 80% of the time and 3 of model B which<br />
are in use 60% of the time. If a manager walks into the factory at a random selected time, what is the probability that<br />
(i) he will find 2 machines of model A and 1 of model B in use, (ii) he finds 2 machines in use.<br />
28. If, over a certain period of time, rain falls at random and on the average on 12 days in every 30, find the<br />
probability that (i) the first 3 days of a given week will be fine and the remainder wet, (ii) rain will fall on just 3 days of<br />
a given week, (iii) at least 3 days in a given week will be fine.<br />
29. A manufacturer of gold chains finds that on the averge 1 gold chain in every 20 is faulty, The gold chains are<br />
marked in packets of 5. Out of a batch of 200 packets, how many would be expected to have 0, 1, 2, 3, 4, 5 faulty<br />
chains.<br />
30. During the course of a particular morning, a machine in a factory workshop was active or idle as specified belows:<br />
Time (AM) Status<br />
9:00-9:30 Idle<br />
9:30-10:20 Active<br />
10:20-10:30 Idle<br />
10:30-11:30 Active<br />
11:30-12:00 Idle<br />
If a work engineer had walked through the workshop at a time randomly selected between 9:00 and 12 noon, determine<br />
the probability that he would have found the machine idle.<br />
Determine also the probability that if he had observed it at six randomly selected times during this period he would<br />
have found it idle (i) at all of them, (ii) at all but one of them.<br />
31. In order to form an estimate of the quality of mass-produced articles, a customer adopts the following double<br />
sampling scheme. A random sample of 10 is inspected; if no defectives are found, he accepts the lot; if 4 or more<br />
defectives are found he rejects the lot; but if 1,2 or 3 defectives are found, he takes a second sample of 10 and if the<br />
total of defectives in the two samples is 4 or more, he rejects the lot, but if the total of defectives in the two samples is 4<br />
or more, he rejects the lot, but otherwise accepts the lot. Calculate the probability that a lot which is known to be 10%<br />
defective is accepted.<br />
32. If p is the probability of a success in each of n independent trials, show, by using the binomial expansion of (q+p)n<br />
that the most likely of successes is np.<br />
33. Find the most likely number of successes in each of the following:<br />
(a) Number of heads in 20 tosses of a fair coin.<br />
(b) Number of defective articles in a sample of 25 articles drawn from a population which is 10% defective.<br />
(c) Number of people in favour of a certain candidate in a sample of 30 voters taken from a population which is 60% in<br />
favour of the candidate.<br />
ANSWERS: 1. 0.08883 2. 0.0002441 3.(i) 0.8192 (ii) 0.0272 4. 21/32 5. (i) 0.8916<br />
(ii) 0.3415 6.(i) 0.3087 (ii) 0.4718 (ii) 0.03087 7.(i) 0.001573 (ii) 0.001678 8. 64/3125<br />
9. 0.04031 10. 0.3174 11.(i) 1/64 (ii) 27/64 (iii) 27/32 (iv) 3/32 12. 0.5155 13. 5/8<br />
14. 1/16 15.(i) 0.9893 (ii) 0.9192 16. 4 17. 11/64 18. 0.394 19. 6 20. 0.0625 21. 0.8824<br />
22. (i) 3471/4096 (ii) (15/16)4 23.(i) 0.6561 (ii) 113/625 (iii) 177/1250 24. 0.0041,<br />
0.0368, 0.2764, 0.3109, 0.1865, 0.0466 (a) 10 (b) 117 25. 19 26. 329 27.(i) 0.04422<br />
(ii) 0 0.0179 28.(i) 0.0055 (ii) 0.2903 (iii) 0.9038 29. 155, 41, 4, 0, 0, 0 30. 7/18;<br />
(i) 0.003457 (ii) 0.0326 31. 0.868 33. (a) 10 (b) 2 (c) 18.<br />
Some Challenge Problems in Probability<br />
1. The Sock Drawer<br />
A drawer contained red socks and black socks. When two socks are drawn at random, the probability that both are red<br />
is 1/2. (a) How small can the number of socks in the drawer be? (b) How small if the number of black socks is even?<br />
2. Successive Wins<br />
To encourage Elmer's promising tennis career, his father offers him a prize if we wins (at least) two tennis sets in a row<br />
in a three- set series to be played with his father and the club champion alternately: father-champion-father or<br />
champion-father-champion, according to Elmer's choice. The champion is a better player than his Elmer's father.<br />
Which series should Elmer choose?<br />
3. Trials Until First Success<br />
On the average, how many times must a die be thrown until one gets a 6?
4. Coin in Square<br />
In a common carnival game, a player tosses a penny from a distance about 5 feet onto the surface of a table rule in 1inch<br />
squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does not<br />
get his penny back; otherwise, he loses his penny. If the penny lands on the table, what is his chance to win?<br />
5. Silent Cooperation<br />
Two strangers are separately asked to choose one of the positive whole numbers and advised that if they choose the<br />
same number, they both get a prize. If you were one of these people, what number would you choose?<br />
6. Quo Vadis<br />
Two strangers who have a private recognition signal agree to meet on a certain Thursday at 12 noon in New York City,<br />
a town familiar to neither, to discuss an important business deal, but later they discover that they have not chosen a<br />
meeting place, and neither can reach the other because both have embarked on trips. If they try nevertheless to meet,<br />
where should they go?<br />
7. Collecting Coupons<br />
Coupons in cereal are numbered 1 to 5, and a set of one of each is required for a prize. With one coupon per box, how<br />
many boxes on the average are required to make a complete set?<br />
8. The Theatre Row<br />
Eight eligible and seven beautiful models happen randomly to have purchased single seats in the same 15-seat row of a<br />
theatre. On the average, how many pairs of adjacent seats are ticketed for marriageable couples?<br />
9. An Even Split at Coin Tossing<br />
When 100 coins are tossed, what is the probability that exactly 50 are heads?<br />
10. Isaac Newton Helps Samuels Pepys<br />
Pepys wrote to Newton to ask which of three events is more likely: that a person get (a) at least 1 sex when 6 dice are<br />
rolled; (b) at least 2 sixes when 12 dice are rolled; (c) at least 3 sixes when 18 dice are rolled. What is the answer?<br />
11. The Ballot Box<br />
In an election, two candidates, Albert and Ben, have in a ballot box a and b votes, respectively, a>b, for example, 3<br />
and 2. If ballots are randomly drawn and tallied, what is the chance that at least once after the first tally the candidates<br />
have the same number of tallies?<br />
12. Players A and B match pennies N times.<br />
They keep a tally of their gains and losses. After the first toss, what is the chance that at no time during the game, will<br />
they be even?<br />
13. Select the Largest Dowry<br />
In order to test a candidate for the position of a wise man, the king offers him a chance to marry the young princess<br />
with the largest dowry. The amounts of dowries are written on slips of paper and mixed. A slip is drawn at random and<br />
the wise man must decide whether that is the largest dowry or not. If he decides it is, he gets the princess and her<br />
dowry, if he is correct; otherwise, he gets nothing. If he decides against the amount written on the first slip, he must<br />
choose or refuse the next slip and son on until he chooses one or else the slips are exhausted. In all 100 attractive<br />
princesses (the king as a lot of wives!) participated, each with a different dowry. How should the wise man make his<br />
decision?<br />
14. Random Quadratic Equations<br />
What is the probability that the quadratic equation x^2 + 2bx + c = 0 has real roots?<br />
15. Broken Bar<br />
A bar is broken at random in two places. Find the average size of the smallest, of the middle-sized, and of the largest<br />
pieces.<br />
16. Thick Coin<br />
How thick should a coin be to have a 1/3 chance of landing on edge ?<br />
17. Birthday problems<br />
(a) What is the least number of persons required if the probability exceeds 1/2 that two or more of them have the same<br />
birthday? (Year of birth need not match.)<br />
(b) You want to find someone whose birthday matches you. What is the least number of strangers whose birthdays you<br />
need to ask about to have a 50-50 chance?<br />
(c) If k persons compare birthdays in the pairing problem, the probability is P(k) that at least 2 have the same birthday.<br />
What should n be in the personal birthmate problem to make your probability of success approximately P(k) ?<br />
18. Prisoner's dilemma<br />
Three prisoners A, B and C, who equally have good records, have applied for parole. The parole board has decided to<br />
release two of the three, and the prisoners know this but not which two. A warder friend of prisoner A knows who are to<br />
be released. Prisoner A realises that it would be unethical to ask the warder if he, A, is to be released, but thinks of<br />
asking for name of ONE prisoner OTHER THAN HIMSELF who is to be released.<br />
He thinks that before he asks, his chance of release are 2/3. He thinks that if the warder say "B will be released", his<br />
own chances have now gone down to 1/2, because either A and B or B and C are to be released. And so A decides not<br />
to reduce his chance by asking. However, A is mistaken in his calculation, why ?<br />
19. The Flippant Juror
There are two jury systems. First, a three-man jury has two members each of whom independently has probability P of<br />
making the correct decision and a third member who is undecided and only makes decision by flipping a coin for each<br />
decision (majority rules). The second system is a one-man jury, which has probability P of making a correct decision.<br />
Which jury has the better probability of making the correct decision?<br />
20. Length of Random Chords<br />
If a chord is selected at random on a fixed circle, what is the probability that its length exceeds the radius of the circle?<br />
21. Catching the Cautious Counterfeiter<br />
(a) The King's minter boxes his coins 100 to a box. In each box, he put 1 false coin. The King suspects the minter and<br />
from each of 100 boxes draws a random coin and has it tested. What is the chance the minter's peculations go<br />
undetected?<br />
(b) What if both 100's are replaced by N?<br />
22. Catching the Greedy Counterfeiter<br />
The King's minter boxes his coins N to a box. Each box contains M false coins. The King suspects the minter draws 1<br />
coin from each of N boxes and has these tested. What is the chance that the sample of N coins contain exactly R false<br />
ones?<br />
23. Evening the Sales<br />
A bread salesman sells on the average 20 cakes on a round of his route. What is the chance that he sells even number<br />
of cakes> (We assume the sales follow the Poisson distribution)<br />
24. Choosing the Largest Random Number<br />
The King wants a wise man to choose the largest number from among 100 numbers. The numbers on slips are<br />
randomly drawn from the numbers from 0 to 1 (random numbers, uniformly distributed). Now, what should the wise<br />
man's strategy be?<br />
25. Random Walk<br />
Starting from an origin O, a particle has a 50-50 chance of moving 1 step north or 1 step south, and also a 50-50<br />
chance of moving 1 step east or 1 step west. After the step is taken, the move repeated from the new position and so on<br />
indefinitely. What is the chance that the particle returns to the origin?<br />
26. Three Dimensional Random Walk<br />
As in the two-dimensional walk, a particle starts at an origin O in three-space. Think of the origin as centered in a<br />
cube 2 units on a side. One move in this walk sends the particle with equal likelihood to one of the eight corners of the<br />
cube. Thus, at every move, the particle has a 50-50 chance of moving one unit up or down, one unit east or west, and<br />
one unit north or south. If the walk continues forever, find the fraction of particles that return to the origin.