Question

Two tennis professionals, A and B, are scheduled to play a match; the winner is the first player to win three sets in a total that cannot exceed five sets. The event that A wins any one set is

P(A) = 0.7,

and is independent of the event that A wins any other set. Let x equal the total number of sets in the match; that is, x = 3, 4, or 5.

(a)

Find

p(x).

x	3	4	5
p(x)

(b)

Find the expected number of sets required to complete the match for P(A) = 0.7.

(c)

Find the expected number of sets required to complete the match when the players are of equal ability—that is, P(A) = 0.5.

(d)

Find the expected number of sets required to complete the match when the players differ greatly in ability—that is, say, P(A) = 0.8.

sets

(e)

What is the relationship between P(A) andE(x), the expected number of sets required to complete the match?

As the probability of winning a single set, P(A), increases from 0.5, the expected number of sets required to complete the match, E(x), increases.

As the probability of winning a single set, P(A), increases from 0.5, the expected number of sets required to complete the match, E(x), decreases.     

As the probability of winning a single set, P(A), increases from 0.5, the expected number of sets required to complete the match, E(x), stays the same.

Answer 1

a)

P(X=3) =P(A wins first 3 )+P(B wins first 3 ) =0.7^3+0.3^3 =0.3700

P(X=4) =P(A wins 2 of first 3 and wins 4 th)+P(B wins 2 of first 3 and wins 4 th) =0.3654

P(X=5)=P(A wins 2 of first 4 and wins 5 th)+P(B wins 2 of first 4 and wins 5 th) =0.2646

x	3	4	5
P(x)	0.37	0.3654	0.2646

b)

Expected number =ΣxP(X) =3*0.37+4*0.3654+5*0.2646 =3.8946

c)

Expected number =ΣxP(X) =3*0.25+4*0.375+5*0.375 =4.125

d)

Expected number =ΣxP(X) =3*0.52+4*0.3264+5*0.1536 =3.6336

e) As the probability of winning a single set, P(A), increases from 0.5, the expected number of sets required to complete the match, E(x), decreases.