Problem 67
Question
Consider a gambler who, at each gamble, either wins or loses her bet with respective probabilities \(p\) and \(1-p .\) A popular gambling system known as the Kelley strategy is to always bet the fraction \(2 p-1\) of your current fortune when \(p>\frac{1}{2}\) Compute the expected fortune after \(n\) gambles of a gambler who starts with \(x\) units and employs the Kelley strategy.
Step-by-Step Solution
Verified Answer
To find the expected fortune of a gambler employing the Kelley strategy after n gambles, starting with x units, and assuming \(p>\frac{1}{2}\), follow these steps:
1. Calculate the fraction of fortune bet: \(2p-1\).
2. Set the initial fortune value: \(EF = x\).
3. Iterate through each gamble from i = 1 to n, computing: \(EF_i = EF_{i-1} \times (p \times (2p-1) + (1-p) \times (1-(2p-1)))\).
4. After n gambles, the final expected fortune will be \(EF_n\).
1Step 1: Restate the given information.
It's given that the gambler follows the Kelley strategy, meaning that they always bet the fraction (2p-1) of their current fortune when the probability of winning (p) is greater than 1/2. The initial fortune is x units.
2Step 2: Bring up the condition for following the Kelley strategy.
The Kelley strategy can only be employed when the probability of winning (p) is greater than 1/2. So, in this exercise, we can assume the condition \(p > \frac{1}{2}\) holds true.
3Step 3: Determine the fraction of fortune bet.
According to the Kelley strategy, the fraction of the current fortune that the gambler should bet is (2p-1).
4Step 4: Find the expected fortune after each gamble.
The expected fortune (EF) after each gamble can be calculated using the formula: \(EF = \text{current fortune} \times (\text{probability of winning} \times \text{fraction bet} + \text{probability of losing} \times (1 - \text{fraction bet}))\)
5Step 5: Calculate the expected fortune after n gambles.
To calculate the expected fortune after n gambles, we need to use the formula found in step 4 and find the result iteratively, as follows:
1. Set the initial fortune value, i.e., \(EF = x\).
2. For each gamble (from i = 1 to n), compute:
\(EF_i = EF_{i-1} \times (p \times (2p-1) + (1-p) \times (1-(2p-1)))\)
3. After n gambles, the final expected fortune will be \(EF_n\).
6Step 6: Conclusion.
By following the Kelley strategy and calculating the expected fortune after each gamble using the formulas above, we can find the expected fortune of the gambler after n gambles, starting with x units.
Other exercises in this chapter
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