← JP Morgan Interview Insights
Pretty clean calculation once you remember expected value is just the probability-weighted average.
Start by clearly defining the random variable and its distribution, then compute the expected value using the formula for a discrete uniform distribution. Finally, interpret the result to determine the maximum rational price to play, emphasizing that paying exactly the expected value yields zero profit in the long run.
Pro tip: Mention that while the expected value is $3.50, risk aversion or utility theory might make a rational person pay less, and in a real-world scenario, the house would charge more than $3.50 to profit—this shows you understand both the math and its practical implications.
Let X be the winnings from one roll. X is uniformly distributed over {1,2,3,4,5,6} with each outcome having probability 1/6.
Calculate E[X] = (1+2+3+4+5+6)/6 = 21/6 = 3.5. This represents the average winnings per play over many repetitions.
The most you should pay is the expected value, $3.50, because paying more would result in a negative expected net gain. Paying exactly $3.50 yields a fair game with zero expected profit.
Acknowledge that in reality, factors like risk aversion, utility of money, and the counterparty's need for profit might influence the decision, but from a purely expected-value standpoint, $3.50 is the threshold.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the base game (e.g., rolling a fair die) and compute its expected value. Then, model the reroll option as a real option: after seeing the first result, you can either keep it or pay a fee to reroll and accept the new outcome. Determine the optimal threshold by comparing the first roll to the expected value of rerolling minus the fee, and compute the overall expected value with the option.
Pro tip: Frame the problem in financial terms: the fee is like an option premium, and the decision rule is a threshold policy. This shows you can connect algorithms to business value, which is highly valued at JP Morgan.
Confirm the rules: what is the random variable (e.g., fair six-sided die), and what is the initial expected value without any reroll option?
After the first roll, you observe a value x. You can keep x or pay a fee f to reroll and accept the new outcome. The optimal decision is to reroll if x < E[reroll] - f, where E[reroll] is the expected value of a fresh roll.
Set the threshold T = E[reroll] - f. Since a reroll gives the original expected value μ, T = μ - f. Reroll if x < T, otherwise keep x.
The expected value with the option is the average over all possible first rolls of the maximum between keeping x and the net value of rerolling (μ - f). This can be computed as (1/n) * sum over x of max(x, μ - f).
The value of the option is the difference between the expected value with the option and the base expected value μ. Discuss how this changes with the fee f and the distribution.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Sum case is easy, linearity of expectation means it's just 3.5 + 3.5 = 7.
Start by clarifying the problem: two fair six-sided dice, and we need the expected value of the sum and then the expected value of the maximum. For the sum, use linearity of expectation. For the maximum, either enumerate all 36 outcomes or use the distribution of the maximum (P(max ≤ k) = (k/6)^2) to compute the expected value.
Pro tip: Mention that linearity of expectation works even if the dice are not independent, but for the maximum, independence is crucial for the distribution calculation. Also, note that the expected maximum is higher than the expected sum divided by 2, which might be counterintuitive.
Confirm that the dice are fair and six-sided, and that we are calculating expected values for two separate scenarios: sum and maximum.
Use linearity of expectation: E[X+Y] = E[X] + E[Y]. Since each die has expected value 3.5, the sum is 7.
Compute the distribution of the maximum: P(max ≤ k) = (k/6)^2 for k=1..6. Then use E[max] = Σ P(max ≥ k) or E[max] = Σ k * P(max = k).
Compute the expected maximum: E[max] = Σ_{k=1}^6 P(max ≥ k) = Σ_{k=1}^6 (1 - ((k-1)/6)^2) = 161/36 ≈ 4.472. Alternatively, enumerate all 36 outcomes to verify.
Note that the expected maximum (≈4.47) is greater than half the expected sum (3.5), which makes sense because the maximum is biased toward higher values.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.