Two teams play an extended series of 100 volleyball games. Team A wins the first game and then team B wins the second. The probability that a team wins each subsequent game is equal to the proportion of games that team has won so far. The probability that one of the teams wins exactly 30 games can be expressed as a/b.What is value of a and b

13

sorry, 13 is exactly what..

is it the value of a/b.
I want a+b.

Thank You for answering

To find the probability that one of the teams wins exactly 30 games, we need to consider the specific sequence of wins and losses that leads to this outcome. Let's break down the problem step by step:

1. After the first game, Team A has won 1 game out of 1, so their winning probability is 1/1 = 1.

2. After the second game, Team A has won 1 game out of 2, so their winning probability is 1/2.

3. For the third game, there are two possible outcomes: Team A wins or Team B wins. If Team A wins, their winning probability becomes 2/3 because they will have won 2 games out of 3.

4. If Team B wins the third game, their winning probability would be 1/3.

5. After the fourth game, there are again two possibilities: Team A wins or Team B wins. If Team A wins, their probability would be 3/4, and if Team B wins, their probability would be 2/4 = 1/2.

By following this pattern, we can observe that the probability of one of the teams winning exactly 30 games depends on the specific sequence of wins and losses. However, we can calculate the probability by considering all possible outcomes:

1. Calculate the total number of possible outcomes by raising 2 to the power of 99, which is the number of remaining games.

Total outcomes = 2^99

2. Calculate the number of favorable outcomes, where one team wins exactly 30 games. Let's assume that it is Team A. The remaining 99 - 30 = 69 games must be divided between Team A and Team B. We can select 69 out of 99 games for Team A and the remaining 30 will be for Team B.

Number of favorable outcomes = C(99, 69)

where C(n, r) represents the number of combinations of choosing r items out of n. It can be calculated using the formula:

C(n, r) = n! / (r! * (n-r)!)

3. Finally, the probability can be calculated as the ratio of the number of favorable outcomes to the total number of outcomes.

Probability = Number of favorable outcomes / Total outcomes

So, to find the values of a and b in the expression a/b, you need to determine the number of favorable outcomes and the total number of outcomes as explained above. Then, a would be the number of favorable outcomes, and b would be the total number of outcomes.