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Calculate Expected Payoff from Two Dice

SIG (Susquehanna)Data Scientist · Online Assessment
EasyData ScientistOnline AssessmentStatistics & Math

Last updated

This problem tests fundamental knowledge of probability, expected value, and partitioning sample spaces to calculate the average outcome of a stochastic process.

The Two-Dice Game

You are playing a game with two fair, six-sided dice. The rules for the payoff are as follows:

  1. If the two dice show different numbers (e.g., a 2 and a 4), your payoff is the maximum of the two numbers.
  2. If the two dice show the same number (a double, e.g., 3 and 3), you lose an amount equal to the sum of the dice. For example, rolling two 3s results in a payoff of -6.

Calculate the expected value of the payoff for a single roll of these two dice. Provide your answer rounded to the nearest cent.

Constraints

  • Each die is independent and has faces labeled 1 through 6.
  • All 36 possible outcomes are equally likely.

Examples

  • If you roll (1, 2), the payoff is max(1, 2) = 2.
  • If you roll (4, 4), the payoff is -(4 + 4) = -8.

Output Format

Provide the expected value as a single decimal number.

Hints

Hint 1

Identify the 36 equally likely outcomes of rolling two dice.

Hint 2

Calculate the payoff for each of the 36 outcomes, distinguishing between cases where the dice are equal and where they are unequal.

Hint 3

Sum all individual payoffs and divide by 36 to find the arithmetic mean.

Follow-up questions

  • How does the expected value change if you only receive a payoff for doubles but lose the maximum of the dice for non-doubles?
  • If we introduce a third die to the game, how would the state space increase and what impact would that have on the complexity of calculating the expectation?
  • Suppose you are allowed to re-roll one die of your choosing after seeing the initial result. What strategy would maximize your expected payoff?