J16

Probability
Probability is a numerical value given to the chance that a certain event will occur.
number of favourabl e outcomes
Probability that an event occurs =
total number of outcomes
Definitions
(i)
(ii)
(iii)
(i)
An experiment is any process by which data is obtained.
The results of an experiment are called outcomes.
A set of all possible outcomes that may occur in a particular experiment is called
a sample space, usually denoted by S. For example,
(a) when a coin is tossed: S = {H, T}
(b) when two coins are tossed: S = {HT, TH, HH, TT}.
(c) When three coins are tossed:
S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}
(d) when a die is thrown: S = {1, 2, 3, 4, 5, 6}, e.t.c.
An event is a set consisting of possible outcomes of an experiment with desired
qualities. It is a subset of a sample space.
Review of terms used in set theory.
1. Intersection of events.
Consider two events A and B of the sample space S, the intersection of these two
events is a set given by E = AB, i.e. containing sample points common to both A
and B.
2. Union of events, denoted by AB, is the set of all sample points in either A or B
or both.
3. Compliment of an event.
If A is an event of a sample space S, the compliment of A, denoted by A, is
given by the set containing all sample points in S that are not in A.
4. Mutually exclusive events
If two events A and B are mutually exclusive, then they cannot occur at the same time.
In other words AB =. For example, selecting ‘an even number’ or selecting a ‘one’
from a set of numbers; In tossing a coin, the events ‘Head’ and ‘Tails’ are mutually
exclusive. If you get a ‘Head’, you cannot get a ‘Tail’ in the same toss.
If a trial (experiment) has a set of equally likely possible outcomes S, then the probability
of the event E occurring is given by

( E)
P (E) =
( S)
Independent events
Two events are independent if the occurrence of one event is unaffected by the
occurrence of the other. E.g. Obtaining a ‘head’ on one coin, and a ‘tail’ on the other
coin when the coins are tossed at the same time.
Venn diagrams
Probabilities can be illustrated on a Venn diagram. The rectangle represents the entire
sample space, and the circle represents the event A, as shown below.
A
When dealing with events, the words and and or are sometimes replaced by the
symbols and respectively. The event of A not happening is sometimes
written as A .
Combinations of probabilities can be shown on a Venn diagram, as follows.
A and B, i.e. AB. This is the overlap of the regions corresponding to A and B as shown
in the figure below.
A or B i.e. AB is the region of points in either the A region or the B region (or both).
Note that the word or is inclusive. ‘A or B’ means ‘A or B or both’, as shown below.

Basic rules of probability
The following are some of the most basic rules of probability.
1. Probabilities are real numbers on the interval from 0 to 1. i.e. 0
P ( E)
1
for any
event E.
2. If an event is certain to occur, its probability is 1, and if an event is certain not to
occur, its probability is 0.
3. If A and B are any two events, then
P(A or B) = P(A) + P(B) – P(A and B).
4 If two events are mutually exclusive, the probability that one or the other will occur
equals the sum of their probabilities
i.e. P (A or B) = P (A) + P (B). This is the addition law for mutually exclusive
events. It is also known as the ‘or rule’
5. The sum of the probabilities that an event will occur and that it will not occur is
equal to 1. i.e. P (A) + P (A) = 1 for any event A,
or P (A) = 1 – P (A).
6. If A and B are independent events then
P(A and B) = P(A) × P(B)
This is the multiplication law for independent events. It is also known as the
‘and rule’
In questions on probability, when the following terms are used, great care must be
taken. To understand the terms we take, for example a family of 4 children, where B
represents a boy and G represents a girl.
(i) at least
‘at least two are boys, B ≥ 2. Therefore, B can take values 2, 3 or 4.
(ii) ‘at most’
‘at most two are boys’, i.e. B 2. Therefore, B can take values 2, 1 or 0.
(iii) Not more than
‘Not more than two boys’ and ‘at most two boys’ mean the same.
(iv) Not less than
‘not less than two boys ‘ and ‘at least two boys’ mean the same.
(v) More than

‘more than two boys’, i.e. B > 2. Therefore B can take values 3 or 4.
(vi) Less than
‘less than two boys’, i.e. B < 2. Therefore, B can take values 0 or 1.
Example
The numbers 1 to 20 are each written on a card. The 20 cards are mixed together. One
card is chosen at random from the pack. Find the probability that the number on the card
is:
(a) even (b) a factor of 24
(c) prime.
Solution
We will use ‘P(x)’ to mean ‘the probability of x’. Let S be the sample space such that S =
{1, 2, 3, 4, 5, 6, 7, 8, ……, 20}
number of even numbers 10 1
(a) P(even) =
= =
total number of numbers in the pack 20 2
number of actors f of 24
(b) P(a factor of 24) =
total number of numbers in the pack
The factors of 24 are: F24 = {1, 2, 3, 4, 6, 8 12, 24}.
∩F ( )
24
= 8
8 2
Therefore, P(a factor of 24) = = 20 5
(c) Prime numbers in the pack = {2, 3, 5, 7, 11, 13, 17, 19 }
number of prime numbers in the pack 8 2
P(prime) =
= = .
total number of numbers in the pack 20 5
Example
A black die and a white die are thrown at the same time. Display all the possible
outcomes. Find the probability of obtaining:
(a) a total of 5,
(b) a total of 11,
(c) a‘two’ on the black die and a ‘six’ on the white die.
Solution
It is convenient to display all the possible outcomes on a grid. There are 36 possible
outcomes, shown here below.
Number on
black die
Number on white die
1 2 3 4 5 6
1 (1,1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
2 (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
3 (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
4 (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)

5 (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
6 (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
(a) There are four ways of obtaining a total of 5 on the two dice. These are:
(1, 4), (4, 1), (2, 3) and (3,2)
Probability of obtaining a total of 5 = 36
4 = 9
1 .
(b) There are two ways of obtaining a total of 11.
P(total of 11) = 36
2 = 18
1
(c) There is only one way of obtaining a ‘two’ on the black die and a ‘six’ on the white
die.
Therefore, P(a two on black and a 6 on white = 36
1
Example
A fair coin is tossed and a fair die is rolled. Find the probability of obtaining a
‘head’ and a ‘six’.
Solution
When a coin is tossed once, the sample space is: S = {H, T}
Where H denotes a ‘head’ and T a ‘tail’. So P(H) = P(T) = ½.
Similarly the sample space when a die is tossed once is: S = {1, 2, 3, 4, 5, 6}
P(six) = 6
1
The two events are independent.
1 1 1
Therefore, P(head and six) = × = .
2 6 12
Exercise
In this exercise, all dice are normal cubic dice with faces numbered 1 to 6.
1. A fair die is thrown once. Find the probability of obtaining:
(a) a six,
(b) an even number,
(c) a number greater than 3,
(d) a three or a five.
2. The two sides of a coin are known as ‘head’ and ‘tail’. Two coins are tossed at the
same time. List all the possible outcomes. Find the probability of obtaining
(a) two heads, (b) a head and a tail.
3. A bag contains 6 red balls and 4 green balls.
(a) Find the probability of selecting at random:
(i) a red ball (ii) a green ball.
(b) One red ball is removed from the bag. Find the new probability of selecting at
random:

(i) a red ball (ii) a green ball.
4. One letter is selected at random from the word ‘UNNECESSARY’. Find the
probability of selecting:
(a) an R (b) an E (c) an O
5. Three coins are tossed at the same time. List all the possible outcomes. Find the
probability of obtaining:
(a) three heads,
(b) two heads and one tail,
(c) no heads,
(d) at least one head.
6. A bag contains 10 red balls, 5 blue balls and 7 green balls. Find the probability of
selecting at random:
(a) a red ball,
(b) a green ball,
(c) a blue or a red ball,
(d) a red or a green ball.
7. A red die and a blue die are thrown at the same time. List all the possible outcomes
in a systematic way. Find the probability of obtaining:
(a) a total of 10.
(b) a total of 12.
(c) a total less than 6.
(d) the same number on both dice.
(e) a total more than 9.
8. A die is thrown; when the result has been recorded, the die is thrown a second time.
Display all the possible outcomes of the two throws. Find the probability of
obtaining:
(a) a total of 4 from the two throws.
(b) a total of 8 from the two throws,
(c) a total between 5 and 9 inclusive from the two throws,
(d) a number on the second throw which is four times the number on the first
throw.
9. One ball is selected at random from a bag containing 12 balls of which x
are white.
(a) What is the probability of selecting a white ball?
When a further 6 white balls are added the probability of selecting a white
ball is doubled.
(a) Find x.
10. A coin is tossed and a die is thrown. Write down the probability of obtaining:
(a) a head on the coin,
(b) an odd number on the die,
(c) a ‘head’ on the coin and an odd number on the die.
11. A ball is selected from a bag containing 3 red balls, 4 black balls and 5 green balls.
The first ball is replaced and a second is selected. Find the probability of obtaining

(a) two red balls, (b) two green balls.
12. The letters of the word ‘INDEPENDENT’ are written on individual cards and the
cards are put into a box. A card is selected at random and then replaced and then a
second card is selected. Find the probability of obtaining:
(a) the letter ‘P’ twice,
(b) the letter ‘E’ twice.
13. The table below shows the number of children in 40 randomly chosen families.
No. of children (x) 0 1 2 3 4 5 Over 5 Total
No. of families (f) 6 8 12 5 3 6 0 40
State the probability that a family chosen at random has:
(a) more than 2 children,
(b) an odd number of children,
(c) P(2 ≤ x < 4).
14 Two tetrahedral dice (4 sided), each with faces labeled 1, 2, 3 and 4 are thrown.
The score is the sum of the two numbers on which the dice land. Find the possibility
space and the probability of each element of the space.
15. Two dice (6 sided), each with faces labeled 1, 2, 3, 4, 5, 6 are thrown. The score is
the difference of the numbers showing on the top faces on two dice. For example,
(5, 2) would score 3, as would (2, 5). Find the possibility space and the probability
of each score.
TREE DIAGRAMS
These are used to:
(i)
(ii)
Example
generate a sample space,
solve certain probability problems.
A bag contains 5 red balls and 3 green balls. A ball is drawn at random and then
replaced. Another ball is drawn. What is the probability that both balls are green?
Solution
r
r
g
g
r
g
*
The branch marked involves the selection of a green ball twice. The probability of this
event is obtained by simply multiplying the fractions on the two branches.

3 3 9
Therefore, P (two green balls) = .
8 8 64
Rules:
(1) When the required outcome is given by a path along the branches of a probability
tree, multiply the probabilities along that path.
(2) When the required outcome is given by more than one path in a probability tree,
add the probabilities resulting from each path.
Example
A bag contains 5 red balls and 3 green balls. A ball is selected at random and not
replaced. A second ball is then selected. Find the probability of selecting:
(a) two green balls
(b) one red ball and one green ball.
3 2 3
(a) P (two green balls) = .
8 7 28
5 3 3 5 15
(b) P (one red, one green) = .
8 7 8 7 28
Example
One ball is selected at random from a bag containing 5 red balls, 2 yellow balls and 4
white balls. Find the probability of selecting a red ball or white ball.
Solution
The two events are exclusive.
P (red ball or white ball) = P (red) + P (white)
5 4 9
= = .
11 11 11
Example
A coin is biased so that it shows ‘heads’ with a probability of 3
2 . The same coin is tossed
three times. Find the probability of obtaining:

(a) two tails on the first two tosses. Ans.
9
1
(b) a head, a tail and a head (in that order) Ans.
27
4
(c) two heads and on tail (in any order). Ans.
9
4
Exercise
1. A bag contains 5 red balls, 3 blue balls and 2 yellow balls. A ball is drawn and not
replaced. A second ball is drawn. Find the probability of drawing
(a) two red balls
(b) one blue ball and one yellow ball,
(c) two yellow balls,
(d) two balls of the same colour.
2 2 1
ANSWERS: (a) ; (b)15 ; (c)
9 45
14 ; (d)
45
2. A six sided die is thrown three times. Draw a tree diagram, showing at each
branch the two events: ‘six’ and ‘not six’. What is the probability of throwing a
total of
(a) three sixes
(b) no sixes
(c) one six
(d) at least one six.
1 125 25 91
ANSWERS: (a) ; (b) ; (c) ; (d) .
216 216 72 216
3. A die has its faces marked 0, 1, 1, 1, 6, and 6. Two of these dice are thrown
together and the total score is recorded. Draw a tree diagram.
(a) How many different totals are possible?
(b) What is the probability of obtaining a total of 7?
1
ANSWERS: (a) 6; (b) 3
4. A coin and a die are tossed. All the possible outcomes are shown in the table
coi
n
die
1 2 3 4 5 6
H H1 H2 H3 H4 H5 H6

T T1 T2 T3 T4 T5 T6
Find:
(a) P(a tail and even number) (b) P(head)
(c) P(1, 2 or 3) (d) P(head and 6)
(e) P(head or 6).
5. Two coins are tossed. Represent the possible outcomes on a probability tree and
find the probability that
(a) both are heads
(b) only one is a head
(c) neither is a head.
6. There are 4 red and 6 yellow counters in a box. One is picked out and replaced,
then a second is picked. Draw a probability tree to show all the possible
outcomes and use it to find
(a) P(red)
(b) P(2 yellow)
(c) P(1 red, 1 yellow).
7. The probability that it will rain in a certain part of the country on any one day is 7
2 .
Use a probability tree to find the probability that out of any two days
(a) both will be dry
(b) only one will be dry
(c) it will rain on both days.
8. A fair, six-sided die is thrown three times. Find
(a) P(three sixes)
(b) P(two sixes)
(c) P(one six)
(d) P(no sixes)
(e) P(at least two sixes)
(f) P(at least one six)
9. A fair, six-sided die is thrown. If event S is throwing a 6 and event N is throwing
another number, draw a probability tree to show the possible outcomes of
throwing the die twice. Find
(a) P(two sixes) (b) P(one six)
(c) P(no sixes)
10. If two dice are thrown together, find the probability that:
(a) they show the same number,
(b) they show different numbers,
(c) the sum of their numbers is 5.
11. Anne plays two games of table tennis against Betty. If the probability of Anne
beating Betty in any one game of table tennis is 5
4 , what is the probability of
Betty:
(a) winning both games,

(b) winning the first game and losing the second one,
(c) winning one of the two games.
12. In a super market, it was found that out of 100 apples in a box, 10 were bad and
could not be sold. Three apple are picked at random from the box (replacing the
previous one each time). What is the probability that:
(a) all the three are good,
(b) none of them is good,
(c) 2 or 3 are bad.
13. If two dice are thrown together and their scores recorded as ordered pairs (1, 1),
(1, 2), e.t.c, construct a table to show all the possible outcomes. Find the
probability that they show:
(a) the same number,
(b) different numbers,
(c) 2 as one of the numbers.
14. A bag contains red beads, blue beads and green beads. The probability that a
bead drawn at random from the bag will be red is 5
3 and that it will be blue is 10
3 .
What will be the probability that it will be green?
If the bag contains a total of 100 beads, determine the number of beads of each
colour.
15. What is the set of possible outcomes if a shilling coin and a die are tossed
together? Calculate the probability of a head and a number greater than 3 turning
up.
16. What is the probability of getting two tails in three throws of a coin?
If the first throw shows a head, what is the probability then?
17. In a class of 30 students, 18 are boys and 12 are girls. A teacher chooses three
students at random. Find the probability that:
(a) all three students are girls,
(b) at least one of the three students is a boy?
18. In a shooting practice, two soldiers Ali and Ben fire shots alternatively at a target.
Ali fires first. The probabilities of Ali and Ben hitting the target with any shot are 3
1
and 5
2 respectively. Using a tree diagram, or otherwise, find the probability that:
(a)
(b)
(c)
they will both miss the target with their first shot,
Ben, in his second round, will be the first to hit the target,
Ben will hit the target at least once with his first three shots.
19. A family of three children can be represented by BGB, where B, G and B
represents boy first, followed by a girl, followed by another boy. Write down all
the eight possible codes fro a family of three children.
Taking P(B) = P(G) = ½, find the probability that a family of three children will
contain:
(a) all three boys,

(b)
(c)
at least one girl,
at most one boy.
20. A bag consists of beads of which 5 are red, 7 are blue and 8 are yellow. Three
beads are to be taken, at random. Calculate the probability that the beads will be:
(a) all yellow,
(b)
(c)
(d)
all three of the same colour
all three of different colour,
all three are not of the same colour.