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 ab where a and b are coprime numbers. What is a+b?
The size of the sample space of the last throw is 20*20=400.
Out of the 400 outcomes, Bret needs those where his throw is 11 or more than Shawn's, namely:
(12,1),(13,1),(14,1)...(20,1)
(13,2),(14,2)...(20,2)
...
(19,8),(20,8)
(20,9)
There are ∑(i), i=1,9 outcomes.
Can you figure out the probability, and a+b?
Please tell the answer. I'm not able to get it.
Thanks.
89
To determine the probability that Bret can come back and win the game, we need to analyze the possible outcomes of the last turn. Since each turn involves rolling a fair 20-sided die, the possible results for each player range from 1 to 20.
Let's consider two scenarios:
Scenario 1: Bret rolls a higher number than Shawn.
In this scenario, Bret's score will increase, while Shawn's score remains the same. If Bret manages to roll a number greater than the difference of 10 points, he will win the game. The probability of this occurring can be calculated by finding the probability of rolling a number greater than 10 on a 20-sided die. Since there are 10 numbers greater than 10 (11, 12, ..., 20), and the die has a total of 20 numbers, the probability is 10/20 = 1/2.
Scenario 2: Bret rolls a lower number than Shawn.
In this scenario, Bret's score will increase, but Shawn's score will also increase. If Bret rolls a number less than the difference of 10 points, he will not be able to win the game. The probability of this occurring can be calculated by finding the probability of rolling a number less than or equal to 10 on a 20-sided die. Since there are 10 numbers less than or equal to 10 (1, 2, ..., 10), and the die has a total of 20 numbers, the probability is 10/20 = 1/2.
Since there are only two possible outcomes for the last turn, and each outcome has a probability of 1/2, the probability of Bret winning the game can be determined as the sum of the probabilities of the two scenarios:
Probability of Bret winning = Probability of Scenario 1 + Probability of Scenario 2
= (1/2) + (1/2)
= 1/2 + 1/2
= 1
= 1/1
So the probability that Bret can come back and win the game is 1/1, which means that there is a 100% chance Bret can win.
Therefore, a = 1 and b = 1. Hence, a + b = 1 + 1 = 2.