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/bwhere a and b are coprime positive integers. What is the value of a+b???
the total games plus one is the answer
2/99 or 101
To find the probability that one of the teams wins exactly 30 games in the extended series, we need to calculate the ratio of the number of favorable outcomes (winning exactly 30 games) to the total number of possible outcomes.
Let's start by analyzing the pattern of wins for each team and create a table to track their progress:
Game: Team A Team B
1 1 0
2 1 1
3 2/3 1/2
4 2/4 2/3
5 3/5 2/4
... ... ...
In each game, the probability of winning for each team is equal to the proportion of games they have won so far.
Now, we can see a pattern emerging. For every game after the first two, the probability for Team A to win is (number of games won by Team A) / (total number of games played so far). Similarly, the probability for Team B to win is (number of games won by Team B) / (total number of games played so far).
To find the number of outcomes where Team A wins exactly 30 games, we need to count the number of ways Team A can win 30 games out of the 100 total games. This can be calculated using combinatorics, specifically the binomial coefficient.
The number of ways to choose 30 games out of 100 total games is calculated as:
C(100, 30) = 100! / (30! * (100-30)!) = 29,045,211,853,700
Next, we calculate the probability of this outcome occurring. To do so, we need to multiply the probability of Team A winning each of the selected 30 games and Team B winning the remaining 70 games.
The probability of one specific outcome occurring is (2/3)^30 * (1/3)^70. However, we have multiple ways to arrange these wins within the 100 games, so we need to consider all possible permutations.
Now, we can calculate the total number of permutations for these outcomes. This can be calculated as:
P(100, 30) = 100! / (30! * (100-30)!) = 884,176,199,373,970,195,454,361,600,000
Finally, we can determine the probability of Team A winning exactly 30 games by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = 29,045,211,853,700 / 884,176,199,373,970,195,454,361,600,000
This fraction can then be simplified, and the numerator (a) and denominator (b) can be expressed as coprime positive integers.
To determine the value of a + b, we need to find the sum of these numerator and denominator values.