Define the expected total payout E as the sum of the first roll plus the expected future payout if the first roll is 4, 5, or 6. Set up a recursive equation E = (1/6)(1+2+3) + (1/6)(4+5+6) + (1/2)E, then solve for E. Alternatively, derive E using an infinite series by conditioning on the number of rolls until stopping.
Pro tip: After solving, briefly verify the result by simulation or by checking that the expected number of rolls is 2, which gives a sanity check on the magnitude of E.
Let X_i be the i-th roll and N be the stopping time (first roll in {1,2,3}). The total payout is S = sum_{i=1}^N X_i. We need E[S].
Condition on the first roll: if it's 1,2,3, stop and payout is that value; if it's 4,5,6, payout is that value plus a fresh expected total. This gives E = (1/6)(1+2+3) + (1/6)(4+5+6) + (1/2)E.
Simplify: E = (6/6) + (15/6) + (1/2)E = 1 + 2.5 + 0.5E = 3.5 + 0.5E. Then 0.5E = 3.5, so E = 7.
Express E as sum_{k=0}^∞ P(N > k) * E[roll | continue]? Actually, use E[S] = sum_{i=1}^∞ E[X_i * I(N ≥ i)]. Since N ≥ i iff first i-1 rolls are in {4,5,6}, P(N ≥ i) = (1/2)^{i-1}. Then E[S] = sum_{i=1}^∞ (1/2)^{i-1} * E[X_i | N ≥ i]? But X_i is independent of N ≥ i? Not exactly, but given N ≥ i, the i-th roll is still uniform? Actually, conditioning on N ≥ i means the first i-1 rolls were 4,5,6, but the i-th roll is independent of that event? Yes, because rolls are independent. So E[X_i | N ≥ i] = E[X_i] = 3.5. Thus E[S] = 3.5 * sum_{i=1}^∞ (1/2)^{i-1} = 3.5 * 2 = 7.
Check that the expected number of rolls is 2, and the average value per roll is 3.5, giving 7. This matches the recursion result.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Set up a recursive equation for the expected total payout E, where E = μ + pE, since with probability p you are forced to roll again and the process restarts. Solve for E to get E = μ / (1 - p), assuming p < 1. Discuss the condition for convergence and interpret the result.
Pro tip: Mention that this is a geometric series and that p must be less than 1 for the expected value to be finite; otherwise the expected payout diverges. This shows you understand the underlying assumptions and can communicate them clearly.
Let E be the expected total payout. Explain that the first roll always gives μ, and then with probability p you get another expected payout E.
Write E = μ + pE. This captures that the total expected payout is the first roll's expected value plus the expected value of the rest of the game, which occurs with probability p.
Rearrange the equation to get E(1 - p) = μ, so E = μ / (1 - p).
State that this holds only if p < 1. If p = 1, the game never ends and the expected payout is infinite (or undefined). If p = 0, E = μ, which makes sense.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the game's rules and the stopping strategy (e.g., when the player chooses to stop). Then compute the expected number of rolls under the optimal strategy and set up the equation: expected payout = expected number of rolls × c, solving for c to make net expected value zero.
Pro tip: Demonstrate that you recognize the optimal stopping problem and that the fair fee depends on the player's strategy; mention that if the player can choose when to stop, the expected number of rolls is not simply the maximum possible but derived from dynamic programming.
Ask or state the rules: what constitutes a roll, what the payout is, and whether the player can stop at any time. Confirm that the player acts optimally to maximize expected net payout.
Determine the threshold or condition under which the player stops rolling. For example, if the payout is the value of the last roll, the player might stop when the roll exceeds a certain value.
Using the optimal strategy, calculate the expected number of rolls (including the first free roll). This may involve solving equations or using dynamic programming.
Let E[N] be the expected number of rolls. The expected net payout is E[payout] - c*(E[N]-1) = 0, since the first roll is free. Solve for c = E[payout] / (E[N]-1).
Compute the numerical value of c and explain its meaning: it is the fee per additional roll that makes the game fair. Discuss sensitivity to assumptions.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.