You are playing in a three-set tennis match, which, as usual, will end as soon a

Question

You are playing in a three-set tennis match, which, as usual, will end as soon as either you or your opponent wins two sets. You are a good player, but sadly out of shape, and estimate that the chances of your winning the first, second, and third sets are respectively 2/3, 1/2 and 1/3 respectively, and that these events are independent. a) What is the probability that you win the match? b) Let X denote the number of the first set that you win. Find the PMF of X c) Let Y denote the number of sets played in the match. Find the PMF of Y.

Explanation / Answer

You can win the match by

Event of winning set 1 = S1

Event of winning set 2 =S2

Event of winning set 3 =S3

P(S1) = 2/3; P(Not S1) = 1-P(S1) = 1-2/3 = 1/3

P(S2) = 1/2; P(Not S2) = 1-P(S2) = 1-1/2 = 1/2

P(S3) = 1/3; P(Not S3) = 1-P(S3) = 1-1/3 = 2/3

a) You can win the match by

W :S1 and S2 or S1andS3 or S2andS3

P(W) = P(S1 and S2) + P(S1 and S3) + P(S2 andS3)= P(S1).P(S2)+P(S1).P(S3) +P(S2).P(S3)

= 2/3 x 1/2 + 2/3x1/3 + 1/2 x 1/3 = 1/3 + 2/9 + 1/6 = (6+4+3)/18=13/18

b)

X : Number of the first set that you win

Possible value of

X = 0 if you lose the first two sets

X = 1 if the winning the first set and losing the match or winning the first set and winning the match

X = 2 if losing the first set and winning the second set

X = 3 ; Not possible if first two are lost you lost the match no need to play the third set

For P(X = 0) = Probability losing the first set and losing the second test = P(Not S1) x P(Not S2) =1/3x1/2 = 1/6

X = 1 if the winning the first set and losing the match or winning the first set and winning the match

P((S1 and S2 )or (S1 and NotS2 and S3) or (S1 and NotS2 and Not S3) )= 2/3x1/2 + 2/3x1/2x1/3+2/3x1/2x2/3 =2/3

X=2 ; lose first set and win second set and win third set or lost first set and win second set and lose third set

=P( notS1 and S2 and S3) or (NotS1 and S2 and notS3)= 1/3*1/2*1/3 + 1/3*1/2*2/3= 1/6

X=0 ; P(X=0) = 1/6

X=1 ; P(X=1) = 2/3

X=2 ; P(X=2) = 1/6

C)

Y : Number of sets played in the match

Possible vaalue of Y =2 or 3

Y=2 ; Set 1 win and Set2 win or Set 1 lose and Set 2 lose

P(Y=2) = P(S1 and S2 or Not S1 and Not S2) = 2/3x1/2 + 1/3x1/2 =1/2

Y=3 : Winning match: Set1 win and Set 2 lose and set 3 win or Set1 lose and set2 win and set 3 win

= (S1 and not S2 and S3) or (NotS1 and S2 and S3)

: Losing the match: (Set 1win and Set2 lose and Set3 lose or Set1 lose and set2 win and set3 lose)

: (S1 and NotS2 and NotS3) or (NotS1 and S2 and NotS3)

P(Y=3) = P( (S1 and not S2 and S3) or (NotS1 and S2 and S3) ) or P(S1 and NotS2 and NotS3) or (NotS1 and S2 and NotS3))=

= 2/3x1/2x1/3 + 1/3x1/2x1/3 + 2/3x1/2x2/3 + 1/3x1/2x2/3 = 1/2

Y=2 ; P(Y=2)=1/2

Y=3 ; P(Y=3) =1/2

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