probability question?

Question

2 teams A and B play a best of 7 series. The probability for each team to win the first game is 1/2; but a team that wins a game gets a boost and its probability to win after a victory increases to 2/3 and the probability of the loosing team decreases to 1/3.

Given that team A loses the first three games to team B, what are the odds taht A will win.

Answer 1

After the first game:
P(A will win 2nd game) = 1/3
P(B will win 2nd game) = 2/3

After the second game:
P(A will win 3rd game) = 1/3
P(B will win 3rd game) = 2/3

After the third game:
P(A will win 4th game) = 1/3
P(B will win 4th game) = 2/3

In order for A to win the tournament, they would have to win the 4th, 5th, 6th, and 7th games.

The corresponding probabilities are:
1/3, 2/3, 2/3, 2/3 (after A wins the fourth game, their probability jumps to 2/3)

Multiply those to find the probability of winning the last four games:

P(A wins the tournament) = 8/81
P(A doesn't win the tournament) = 1 - 8/81 = 73/81

Odds that A will win the tournament are 8:73.

*Remember, odds are different than probability.

Answer 2

The best way to approach this is through event tree analysis. These are the feasible pathways (branches) starting with the three losses (L = a win for B) and the joint probabilities along each path. S = a win for A (a success).

LLLSSSS= (1) 1/3 2/3 2/3 2/3 = probability of a series win for A (four S's for team A)
LLLSSSL= (1) 1/3 2/3 2/3 1/3 = series lost
LLLSSL= (1) 1/3 2/3 1/3 = series lost
LLLSL= (1) 1/3 1/3 = series lost
LLLL= (1) 2/3 = series lost

Note: (1) = given that the first three games were lost by A; therefore the probability of losing the first three games = 1.00, it's a fact...a done deal...not an uncertainty at all.

Thus, we can see only one path that ends in a series win for A. Multiply the individual probabilities in the first row and that'll be your answer. Note that branches terminate as soon as the fourth L shows up because B won the series at that point.

Answer 3

Restate the question as follows:

What is the probability that team B will lose the next 4 games? This is what would need to happen for team A to win the best of 7 series.

Prob(B loses game 4) = 1/3
Prob(B loses game 5) = 2/3 because they lost game 4.
Prob(B loses game 6) = 2/3
Prob(B loses game 7) = 2/3

Prob of losing all 4 games:
= 1/3 * 2/3 * 2/3 * 2/3 = 8/81.