Introduction to Probability and Statistics for Engineers and Scientists

3.8Independent Events 79

A

1

2

3

n

B

FIGURE 3.7 Parallel system: functions if current flows from A to B.

It is sometimes the case that the probability experiment under consideration consists of
performing a sequence of subexperiments. For instance, if the experiment consists of
continually tossing a coin, then we may think of each toss as being a subexperiment. In
many cases it is reasonable to assume that the outcomes of any group of the subexperiments
have no effect on the probabilities of the outcomes of the other subexperiments. If such is
the case, then we say that the subexperiments are independent.

EXAMPLE 3.8d A system composed ofnseparate components is said to be a parallel system
if it functions when at least one of the components functions. (See Figure 3.7.) For such
a system, if componenti, independent of other components, functions with probability
pi,i=1,...,n, what is the probability the system functions?

SOLUTION LetAidenote the event that componentifunctions. Then

P{system functions}= 1 −P{system does not function}

= 1 −P{all components do not function}

= 1 −P

(

Ac 1 Ac 2 ···Acn

)

= 1 −

∏n

i= 1

(1−pi) by independence ■

EXAMPLE 3.8e A set ofkcoupons, each of which is independently a typejcoupon with
probabilitypj,

∑n
j= 1 pj = 1, is collected. Find the probability that the set contains
a typejcoupon given that it contains a typei,i=j.

SOLUTION LetArbe the event that the set contains a typercoupon. Then

P(Aj|Ai)=

P(AjAi)

P(Ai)

To computeP(Ai) andP(AjAi), consider the probability of their complements:

P(Ai)= 1 −P(Aci)

= 1 −P{no coupon is typei}

= 1 −(1−pi)k