Introduction to Probability and Statistics for Engineers and Scientists

142 Chapter 5: Special Random Variables

The probability mass function of a binomial random variable with parametersnandp
is given by

P{X=i}=

(

n

i

)

pi(1−p)n−i, i=0, 1,...,n (5.1.2)

where

(n

i

)
=n!/[i!(n−i)!]is the number of different groups ofiobjects that can be
chosen from a set ofnobjects. The validity of Equation 5.1.2 may be verified by first
noting that the probability of any particular sequence of thenoutcomes containingi
successes andn−ifailures is, by the assumed independence of trials,pi(1−p)n−i.
Equation 5.1.2 then follows since there are

(n

i

)
different sequences of thenoutcomes
leading toisuccesses andn−ifailures — which can perhaps most easily be seen by
noting that there are

(n

i

)

different selections of theitrials that result in successes. For

instance, ifn=5,i=2, then there are

( 5

2

)
choices of the two trials that are to result in
successes — namely, any of the outcomes

(s,s,f,f,f)(f,s,s,f,f)(f,f,s,f,s)

(s,f,s,f,f)(f,s,f,s,f)

(s,f,f,s,f)(f,s,f,f,s)(f,f,f,s,s)

(s,f,f,f,s)(f,f,s,s,f)

where the outcome (f,s,f,s,f) means, for instance, that the two successes appeared on
trials 2 and 4. Since each of the

( 5

2

)
outcomes has probabilityp^2 (1−p)^3 , we see that the
probability of a total of 2 successes in 5 independent trials is

( 5

2

)
p^2 (1−p)^3. Note that, by
the binomial theorem, the probabilities sum to 1, that is,

∑∞

i= 0

p(i)=

∑n

i= 0

(n

i

)

pi(1−p)n−i=[p+(1−p)]n= 1

The probability mass function of three binomial random variables with respective param-
eters (10, .5), (10, .3), and (10, .6) are presented in Figure 5.1. The first of these is
symmetric about the value .5, whereas the second is somewhat weighted, orskewed,to
lower values and the third to higher values.

EXAMPLE 5.1a It is known that disks produced by a certain company will be defective
with probability .01 independently of each other. The company sells the disks in packages
of 10 and offers a money-back guarantee that at most 1 of the 10 disks is defective.
What proportion of packages is returned? If someone buys three packages, what is the
probability that exactly one of them will be returned?

SOLUTION IfXis the number of defective disks in a package, then assuming that customers
always take advantage of the guarantee, it follows thatXis a binomial random variable