Data Science Interview Prep: Q63
How Much Would You Pay to Play This Two-Roll Dice Game? (Category: Probability)
Imagine you’re playing a game where you roll a fair six-sided die. You get up to two rolls: after the first roll, you can choose to either keep the number rolled or roll again one last time. Your final payout is the number showing on the die when you stop. How much should you be willing to pay to play this game?
Solution:
To find the fair price to pay for this game, we calculate its expected value assuming you play optimally.
1. Understand the expected value of a single roll.
The average value when rolling a fair six-sided die = (1 + 2 + 3 + 4 + 5 + 6)/ 6 = 3.5.
2. Determining the optimal choice after the first roll.
If your first roll is 1, 2, or 3, the value is less than 3.5, so you should re-roll to try for a better outcome.
If your first roll is 4, 5, or 6, it’s better to keep that roll because it is greater than the average expected value of re-rolling. (Note: When you re-roll, giving up your first roll, you are again looking at an expected value of 3.5 from the second roll.)
3. Calculate the overall expected value.
Scenario 1: Rolling 1, 2 or 3 on the first roll.
Probability of rolling 1, 2, or 3 on the first roll is 3/6 = 0.5. If this happens, you re-roll, which has an expected value of 3.5.
Weighted contribution for this scenario = 0.5 × 3.5 = $1.75.
Scenario 2: Rolling 4, 5 or 6 on the first roll.
Probability of rolling 4, 5, or 6 on the first roll is 3/6 = 0.5. If this happens, you keep that roll. The average of 4, 5, and 6 = (4 + 5 + 6)/3 = 5.
Weighted contribution for this scenario = 0.5 × 5 = $2.5.
Combining both scenarios.
The expected payoff of the overall game = $1.75 + $2.5 = $4.25.
Therefore, you should be willing to pay up to $4.25 to play this game.
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Reference:
Dice Game Optimal Stopping Strategy: https://math.stackexchange.com/questions/1472308/optimal-stopping-with-dice-rolls