Question

Two groups are having competition for chase. If one plays it in its own campus it will win with probability P> ½. They will play three times, two in group 1’s campus and one in group 2’s campus. If one wins game 1 and 2, then game 3 is not going to be played. a) What is P[T], the probability that the group 1 wins the series? b) If they play only once will group 1 have higher chance of winning than from playing 3 games?

Answer 1

Given,

Two groups are having competition for chase. If one plays it in its own campus it will win with probability P > ½.

Probability of Group 1 winning in its own campus = P

Probability of Group 2 winning in Group 1’s campus or Group 1 losing in its own campus = 1 - P

Probability of Group 1 winning in Group 2’s Campus = 1 – P

Answer to 1st Question :
There are 2 cases –

· Case I: Group 1 winning both the games in its own campus

Probability of Group 1 winning both the games in its own campus = P^2

· Case II: Group 1 losing one of the two games in its own campus and winning the third one in Group 2’s campus

Probability of Group 1 winning 1 and losing 1 game = (2C1)xPx(1 - P) = 2P(1 - P)

Probability of Group 1 winning in Group 2’s Campus = 1 – P

The required Probability = 2P(1 - P)(1 - P) = 2P{(1 - P)}^2

Therefore,

Total Probability of Group 1 winning the series, P(T) = (P^2) + 2P{(1 - P)}^2

Answer to 2nd Question:
Since, it is already given, P > ½

Let P = 0.6

If 3 games are played, P(T) = (P^2) + 2P{(1 - P)}^2 = 0.552

Which states that there is 55% chance of Group 1 winning if 3 games are played

If only 1 game is played, there will be two cases –

· Case I: If the game is played in Group 1’s campus

Probability of Group 1 winning = P = 0.6

That is, there is 60% chance of Group 1 winning.

· Case II: If the game is played in Group 2’s campus

Probability of Group 1 winning = 1 - P = 0.4

That is, there is 40% chance of Group 1 winning.

Hence, the chance of winning of Group 1 depends on which campus they are playing

If they are playing in their own campus then they have a better chance of winning then they playing all three games.