In: Statistics and Probability
Two snooker players, Player A and Player B, play a match of a ‘best-of-seven’ frames of snooker, i.e. the first player to win 4 frames wins the match. The probability Player A wins a frame is 0.55 and the probability Player B wins a frame is therefore 0.45.
(a) What is the probability that Player A wins the match by a score
of 4-2?
(b) What is the overall probability Player B wins the match?
(c) What is the expected number of frames played in a match?
P(Chances of A wininng a frame) = P(A) = 0.55
P(Chances of B winning a frame) = P(B) = 0.45
(a) P(Player A wins the match by a score of 4-2) that means from starting 5 matches, A should win 3 matches and 6th match should be won by A
P(Player A wins the match by a score of 4-2) = 5C2 (0.55)3 (0.45)2 * 0.55
= 0.1853
(b) Probability that player B will win this Match =
P(B win intiatil 4 matches) + P(B win 4 out of 5 starting matches) + P(B win 4 out of 6 starting matches) + P(B win 4 out of 7 matches)
= 0.454 + 4C3 (0.55)1 (0.45)3 * 0.45 + 5C3 (0.55)2 (0.45)3 * 0.45 + 6C3 (0.55)3 (0.45)3 * 0.45
= 0.45^4 [ 1 + 4 * 0.55 + 10 * 0.552 + 20 * 0.553]
= 0.3917
(c) Here there are 4 possibilites of number of frames i.e. 4,5,6 and 7
if x is the number of frames to be played than
P(x = 4) = P(A win in 4 straight frames) + P(B win in 4 straight frames)
= 0.554 + 0.454 = 0.1325
P(x = 5) = P(A win in 5 frames) + P(B win in 5 frames)
= 4C3 (0.55)3 (0.45)1 * 0.55 + 4C3 (0.55)1 (0.45)3 * 0.45 = 0.2549
P(x = 6) = P(A win in 6 frames) + P(B win in 6 frames)
= 5C3 (0.55)3 (0.45)2 * 0.55 + 5C3 (0.55)2 (0.45)3 * 0.45= 0.3093
P(x = 7) = P(A win in 7 frames) + P(B win in 7 frames)
= 6C3 (0.55)3 (0.45)3 * 0.55 + 6C3 (0.55)3 (0.45)3 * 0.45= 0.3032
so here
E[X] =
= 4 * 0.1325 + 5 * 0.2549 + 6 * 0.3093 + 7 * 0.3032
= 5.78