When Mr. Lucky starts betting, he has 3 dollars. On any bet, he wins with probability 1/3 and loses with probability 2/3. If he wins
a bet, the total amount of money he has triples. If he loses a bet, he
loses 2/3 of the total amount of money he has. Mr. Lucky’s objective
is to walk away with 27 dollars in his pocket, and he can keep playing as long as he has at least 3 dollars. What is the probability that he reaches his objective? Express your answer as a common fraction.
math - Steve, Tuesday, July 5, 2016 at 4:41am
1/10 to end up with exactly 27
6/10 to end up with at least 27
4/10 to lose
The answer is 1/7, but I don't know how to get it.
Hmmm. I drew a tree with all possible outcomes, but I forgot to weight each branch with its probability. Try doing that and see what happens. I'll try to get back later.
To find the probability that Mr. Lucky reaches his objective of having 27 dollars, we can use a recursive approach. Let's denote the probability of reaching the objective starting with n dollars as P(n).
First, let's establish some base cases:
P(27) = 1, because if Mr. Lucky already has 27 dollars, he has reached his objective.
P(n) = 0 for n < 3, because he cannot reach his objective with less than 3 dollars.
Now, let's consider the cases when Mr. Lucky has more than 27 dollars. In these cases, he can only get to 27 dollars if he loses money. Let's say he has n dollars, where 28 ≤ n < 54. On the next bet, he loses 2/3 of his total money. So, the probability of reaching 27 dollars starting with n is equal to the probability of losing money on the next bet (2/3) multiplied by the probability of reaching 27 dollars starting with 3(n/3), because he will have 2/3 of his current money after the loss.
P(n) = (2/3) * P(n/3), for 28 ≤ n < 54
Now, let's consider the cases when Mr. Lucky has less than 27 dollars. In these cases, he can either win money or lose money. If he wins, he triples his money, so he can reach 27 dollars by starting with 1/3 of the required amount, i.e., 9 dollars. If he loses, he loses 2/3 of his total money. So, the probability of reaching 27 dollars starting with n is equal to the probability of winning money on the next bet (1/3) multiplied by the probability of reaching 27 dollars starting with 3n, plus the probability of losing money (2/3) multiplied by the probability of reaching 27 dollars starting with n/3.
P(n) = (1/3) * P(3n) + (2/3) * P(n/3), for 3 ≤ n < 27
Using this recursive formula, we can calculate P(n) for n in the range of interest, from 3 up to 27. Since we already established P(27) = 1, we can work our way backward to find P(3).
P(27) = 1
P(27/3) = P(9) = (1/3)*P(3*(27/3)) + (2/3)*P(9/3) = 1/3*P(27) +2/3*P(3)
P(9) = 1/3*1 + 2/3*P(3)
P(3) = 1/3*1 - 2/3*P(3)
P(3) + 2/3*P(3) = 1/3
5/3*P(3) = 1/3
P(3) = 1/5
So, the probability that Mr. Lucky reaches his objective of having 27 dollars starting with 3 dollars is 1/5.