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.
math-steve - Steve, Tuesday, July 5, 2016 at 7:43pm
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.
I'm not exactly sure how to weigh it.
To find the probability that Mr. Lucky reaches his objective of 27 dollars, we can use a tree diagram to visualize all the possible outcomes and their probabilities.
Let's start by drawing a tree diagram to represent Mr. Lucky's betting choices.
S (Starting with 3 dollars)
/ \
Win Lose
/ \ / \
9 2 2 1 (After the first bet)
/ \ / \
Win Lose Win Lose
/ \ / \ / \
27 6 4 1 6 0 (After the second bet)
In the first level of the tree, Mr. Lucky can either win or lose the first bet. The probability of winning the first bet is 1/3, and the probability of losing is 2/3.
In the second level, if Mr. Lucky wins the first bet, his money triples to 9 dollars, and if he loses, he loses 2/3 of his money (2 dollars).
In the third level, the same pattern repeats. If Mr. Lucky wins the second bet, his money triples again to 27 dollars, and if he loses, he loses 2/3 of his remaining money.
Now, to find the probability of reaching the objective of 27 dollars, we need to calculate the probability of following the path that leads to 27 dollars in the tree diagram.
The path that leads to 27 dollars is: Start -> Win -> Win -> Win
To find the probability of this path, we multiply the probabilities of each individual step along the path.
Probability of winning the first bet = 1/3
Probability of winning the second bet = 1/3
Probability of winning the third bet = 1/3
Therefore, the probability of reaching the objective of 27 dollars is (1/3) * (1/3) * (1/3) = 1/27.
However, this is just the probability of reaching exactly 27 dollars. Since Mr. Lucky can keep playing as long as he has at least 3 dollars, we need to consider all the paths that lead to having 27 dollars or more.
Looking at the tree diagram, we can see that there are three possible paths that lead to having at least 27 dollars: Start -> Win -> Win -> Win, Start -> Win -> Win -> Lose -> Win, and Start -> Win -> Lose -> Win -> Win.
To find the probability of reaching at least 27 dollars, we add up the probabilities of these three paths:
(1/3) * (1/3) * (1/3) + (1/3) * (1/3) * (2/3) * (1/3) + (1/3) * (2/3) * (1/3) * (1/3) = 1/27 + 2/81 + 2/81 = 5/81.
Therefore, the probability that Mr. Lucky reaches his objective of 27 dollars is 5/81, which is equal to 1/7 (since 5/81 can be simplified).