When Mr. Lucky starts betting, he has 3 dollars. On any bet, he 7. 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.
1/10 to end up with exactly 27
6/10 to end up with at least 27
4/10 to lose
To find the probability that Mr. Lucky reaches his objective of having 27 dollars, we can use a recursive approach to calculate the probabilities of reaching each possible amount of money.
Let's define a function P(x) to represent the probability of reaching the objective starting with x dollars. We can start by setting P(27) = 1 since Mr. Lucky already has the desired amount of money.
Now, let's calculate the probabilities for smaller amounts of money. If Mr. Lucky has less than 27 dollars, he has two options: he can either win a bet and triple his money, or lose a bet and lose 2/3 of his money.
For example, if Mr. Lucky has 26 dollars, he can reach his objective by winning a bet (probability 1/3) and tripling his money to 78 dollars. So, we have:
P(26) = (1/3) * P(78)
Similarly, if Mr. Lucky has 25 dollars, he can either win and reach 75 dollars or lose and have 16.66 dollars (rounded to the nearest cent). So we have:
P(25) = (1/3) * P(75) + (2/3) * P(16.66)
By applying this recursive approach, we can calculate the probabilities for all possible amounts of money until we reach 3 dollars. Finally, we can use these probabilities to find the probability of reaching the objective:
P(3) = (1/3) * P(9) + (2/3) * P(2)
Therefore, the probability that Mr. Lucky reaches his objective of having 27 dollars can be found by evaluating the function P(3).