Bret and Shawn are playing a game where each turn the players roll a 20-sided die fair and then add the number they get to their score. The winner is the player with the highest score at the end of the game (there may be a tie). Before the last turn, Shawn is winning by 10 points. The probability that Bret can come back and win the game can be expressed as a/b where a and b are coprime numbers. What is a+b?

Question

Bret and Shawn are playing a game where each turn the players roll a 20-sided die fair and then add the number they get to their score. The winner is the player with the highest score at the end of the game (there may be a tie). Before the last turn, Shawn is winning by 10 points. The probability that Bret can come back and win the game can be expressed as a/b where a and b are coprime numbers. What is a+b?

Answer 1

To solve this problem, we need to calculate the probability that Bret can come back and win the game, given that he is currently 10 points behind Shawn.

First, let's assume that both Bret and Shawn will make one final roll of the 20-sided die. Bret needs to roll a number higher than Shawn's final roll to win the game. Since the die is fair, there are 20 possible outcomes for each roll, ranging from 1 to 20.

To determine the probability of Bret winning, we need to calculate the probability of him rolling a number higher than Shawn's final roll. Let's consider the different scenarios:

1. Bret rolls a 1: In this case, he cannot win, regardless of Shawn's roll.
2. Bret rolls a 2: He can only win if Shawn rolls a 1. The probability of this happening is 1/20.
3. Bret rolls a 3: He can only win if Shawn rolls a 1 or 2. The probability of this happening is 2/20 since there are two favorable outcomes (1 or 2).
4. Bret rolls a 4: He can only win if Shawn rolls a 1, 2, or 3. The probability of this happening is 3/20.
5. Bret rolls a 5: He can only win if Shawn rolls a 1, 2, 3, or 4. The probability of this happening is 4/20.

We can observe a pattern here where the probability of Bret winning for each roll is equal to the number on the die minus one divided by 20.

Therefore, the probability of Bret coming back and winning is:

(1/20) + (2/20) + (3/20) + ... + (20/20)

To simplify this expression, we can use the formula for the sum of the first n natural numbers:

1/20 + 2/20 + 3/20 + ... + 20/20 = (20 * 21 / 2) / 20 = 21/2 = 10.5/1

So, the probability that Bret can come back and win the game is 10.5/20, which can be expressed as a fraction in the form a/b with a = 21 and b = 2.

Finally, a + b = 21 + 2 = 23. Therefore, the answer is 23.

89

To solve this problem, we need to calculate the probability that Bret can come back and win the game, given that he is currently 10 points behind Shawn.