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 ahead 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?

Details and assumptions:
~Both Shawn and Bret have 1 more roll on the last turn.

~If there is a tie, no one won.

This problem is not about geometry

You want the probability that Bret's last toss will exceed Shawn's by 11 or more.
There are 20x20 = 400 possible outcomes of the two tosses. Here are some of the ways Bret can win ion the last toss.
Bret can throw 12 while Shawn throws 1.
Bret can throw 13 while Shawn throws 1 or 2.
Bret can throw 14 while Shawn throws 1, 2, or 3.
.. (etc) Finally
Bret can throw 20 while Shawn throws 1, 2, 3, 4, 5, 6, 7, 8 or 9.
The total number of possibilities out of 200 that result in Bret winning is
1 + 2 + 3 + 4 + 5 + 6 + 7+ 8 +9 = 45.
The probability of Bret winning is 45/200 = 9/40 (using coprime numbers)
Therefore a = 9 and b = 40.
a + b = 49

answer is 89

Well, it sounds like Bret needs a little extra help to win this game. Let's see if he has a chance!

Since each turn consists of rolling a fair 20-sided die, there are 20 possible outcomes, ranging from 1 to 20.

Now, let's consider the scenario where Bret makes a comeback to win. In order for this to happen, Bret needs to score higher than Shawn in both of their remaining turns.

Let's look at the possibilities step by step:

1. Bret rolls a 1 in the first turn, then rolls x in the second turn where x > 10 (Shawn's current lead). The probability of Bret rolling a 1 is 1/20, and the probability of rolling a number greater than 10 is 10/20. So the probability of this happening is (1/20) × (10/20).

2. Bret rolls a 2 in the first turn, then rolls x in the second turn where x > 9 (since he needs to catch up by 10). The probability of rolling a 2 is 1/20, and the probability of rolling a number greater than 9 is 11/20. So the probability of this happening is (1/20) × (11/20).

3. Continuing this pattern, we can consider all possible outcomes for Bret's first roll, ranging from 1 to 20, and calculate the probability for each.

To find the overall probability of Bret winning, we need to sum up the probabilities of all the favorable outcomes:

(1/20) × (10/20) + (1/20) × (11/20) + ... + (1/20) × (20/20)

Simplifying this equation, we get:

(1/20) × [(10 + 11 + ... + 20)/20] = (1/20) × (10 + 11 + ... + 20) / 20

Now, let's calculate the sum of the numbers from 10 to 20:

(10 + 11 + ... + 20) = (20 × (20 + 10)) / 2 = 330

Plugging this back into the equation, we have:

(1/20) × (330/20) = 33/400

Therefore, the probability that Bret can come back and win the game is 33/400.

Finally, a + b = 33 + 400 = 433.

So, the value of a + b is 433.

To solve this problem, we need to consider all possible outcomes based on the last turn of the game. Let's break it down step by step.

Step 1: Calculate the probability of each possible outcome.

Since both Shawn and Bret have 1 more roll on the last turn, they have a range of possible scores they can achieve. In this case, since they are rolling a 20-sided die, the possible scores they can get on their turn are integers ranging from 1 to 20.

For Shawn to maintain his lead or extend it, he needs to roll a higher number than Bret. So let's consider each possible outcome for Shawn's roll and calculate the probability of each outcome:

- Shawn rolls a 1: The probability of this outcome is 1/20.
- Shawn rolls a 2: The probability of this outcome is 1/20.
- Shawn rolls a 3: The probability of this outcome is 1/20.
- ...
- Shawn rolls a 20: The probability of this outcome is 1/20.

Step 2: Calculate the probability of Bret winning given each possible outcome.

For Bret to come back and win the game, he needs to not only roll a higher number than Shawn but also have a higher total score overall. Let's calculate the probability of each possible outcome where Bret wins:

- If Shawn rolls a 1 and Bret rolls any number higher than 11, Bret wins. The probability of this outcome is (10/20) * (10/20), since both players have an equal chance of rolling any number from 12 to 20 on their turn.
- If Shawn rolls a 2 and Bret rolls any number higher than 12, Bret wins.
- If Shawn rolls a 3 and Bret rolls any number higher than 13, Bret wins.
- ...
- If Shawn rolls a 20, Bret has no chance of winning since his maximum possible score is 20, which is not enough to overcome the 10-point lead.

Step 3: Calculate the overall probability of Bret winning.

To calculate the overall probability of Bret winning, we need to sum up the probabilities of each possible outcome where Bret wins. Since Shawn's roll is independent of Bret's roll, we can simply add up the probabilities from step 2:

(10/20) * (10/20) + (10/20) * (11/20) + (10/20) * (12/20) + ... + (10/20) * (19/20)

Step 4: Simplify the expression and calculate a + b.

We can factor out (10/20) from each term in the sum:

(10/20)(10/20 + 11/20 + 12/20 + ... + 19/20)

Simplifying the inner sum:

(10/20)(170/20)

Simplifying further:

(1/4)(17/2) = 17/8

So a = 17 and b = 8.

Therefore, a + b = 17 + 8 = 25.

The total number of possibilities out of 400

The total number of possibilities out of 400 that result in Bret winning is

1 + 2 + 3 + 4 + 5 + 6 + 7+ 8 +9 = 45.
The probability of Bret winning is 45/400 = 9/80 (using coprime numbers)
Therefore a = 9 and b = 80.
a + b = 89