082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

29.10 The binomial distribution 953
EXERCISES29.9
1 Evaluate
(a) P(8,6)
( ) 12
(c)
9
(b) P(11,7)
( ) 15
(d)
12
2 Writeoutexplicitly
( ( n n
(a) (b)
0)
1)
(c)
(e)
( ) 15
3
( n
2)
3 Theexpansionof (a+b) n wherenisapositiveinteger
maybe written with the help ofcombination notation.
(
(a+b) n =a n n
+ a
1)
n−1 b
( n
+ a
2)
n−2 b 2 +···
( ) ( n
+ ab n−1 n
+ b
n −1 n)
n
n∑
( n
= a
k)
n−k b k
k=0
Expand
(a) (a +b) 4 (b) (1 +x) 6 (c) (p+q) 5
4 Primary and secondary routesconnecting two
computers need to be chosen. Two primaryroutesare
needed from eightwhich are suitable and three
secondary routesmust be chosen from four available.
Inhow many ways can the routesbe chosen?
5 A combination lock can be opened by diallingthree
correct lettersfollowed bythree correct digits. How
many different possibilitiesare there forarranging the
lettersand digits? Isthismore secure than alock
which has seven digits? Isthe word ‘combination’
being usedcorrectly?
6 A nuclear power station isto be builton oneof20
possiblesites.Ateamofengineersiscommissionedto
examine the sites andrank the three most favourable
in order. Inhow many ways can thisbe done?
Solutions
1 (a) 20160 (b) 1663200 (c) 220 (c)p 5 +5p 4 q+10p 3 q 2 +10p 2 q 3 +5pq 4 +q 5
(d) 455 (e) 455
4 112
n(n −1)
2 (a) 1 (b) n (c)
2 5 17576000. Yes,itis more secure. Combination is not
3 (a)a 4 +4a 3 b+6a 2 b 2 +4ab 3 +b 4
being usedcorrectly:permutation lockwould be
better.
(b)1 +6x +15x 2 +20x 3 +15x 4 +6x 5 +x 6 6 6840
29.10 THEBINOMIALDISTRIBUTION
In a single trial or experiment a particular result may or may not be obtained. For example,
in an examination, a student may pass or fail; when testing a component it may
workornotwork.Theimportantpointisthatthetwooutcomesarecomplementary.Assumingthattheprobabilityofanoutcomeisfixed,suchatrialiscalledaBernoullitrial
inhonour ofthe mathematician, J.Bernoulli.
Weaddressthefollowingproblem.InasingletrialtheoutcomeiseitherAorB,thatis
A.WerefertoAasasuccessandBasafailure.IfweknowP(A) = pthenP(B) = 1−p.
Ifnindependenttrialsareobserved,whatistheprobabilitythatAoccursktimes,andB
occursn −k times? The number of successful trials innsuch experiments is a discrete
randomvariable;suchavariableissaidtohaveabinomialdistribution.Letusconsider
a particular problem.