Introduction to Probability and Statistics for Engineers and Scientists

82 Chapter 3:Elements of Probability

10.Show that ifE⊂FthenP(E) ≤P(F). (Hint:WriteFas the union of two

mutually exclusive events, one of them beingE.)

11.Prove Boole’s inequality, namely that

P

(n

⋃

i= 1

Ei

)

≤

∑n

i= 1

P(Ei)

12.IfP(E) = .9 andP(F) = .9, show thatP(EF)≥.8. In general, prove

Bonferroni’s inequality, namely that

P(EF)≥P(E)+P(F)− 1

13.Prove that

(a) P(EFc)=P(E)−P(EF)

(b) P(EcFc)= 1 −P(E)−P(F)+P(EF)

14.Show that the probability that exactly one of the eventsEorFoccurs is equal to

P(E)+P(F)− 2 P(EF).

15.Calculate

( 9

3

)

,

( 9

6

)

,

( 7

2

)

,

( 7

5

)

,

( 10

7

)

.

16.Show that

(

n

r

)

=

(

n

n−r

)

Now present a combinatorial argument for the foregoing by explaining why a

choice ofritems from a set of sizenis equivalent to a choice ofn−ritems from

that set.

17.Show that

(

n

r

)

=

(

n− 1

r− 1

)

+

(

n− 1

r

)

For a combinatorial argument, consider a set ofnitems and fix attention on one

of these items. How many different sets of sizercontain this item, and how many

do not?

18.A group of 5 boys and 10 girls is lined up in random order — that is, each of the

15! permutations is assumed to be equally likely.

(a) What is the probability that the person in the 4th position is a boy?

(b) What about the person in the 12th position?

(c) What is the probability that a particular boy is in the 3rd position?

19.Consider a set of 23 unrelated people. Because each pair of people shares the same

birthday with probability 1/365, and there are

( 23

2

)

=253 pairs, why isn’t the

probability that at least two people have the same birthday equal to 253/365?