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Find the Probability That the Three Smallest Dice Sum to Three

OptiverSoftware Engineer · Technical Screen
HardSoftware EngineerTechnical ScreenStatistics & Math

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This problem tests combinatorics and probability theory by requiring the derivation of the exact probability that the three smallest values of four independent dice rolls sum to exactly three.

The Problem

You are given four fair six-sided dice (D1, D2, D3, D4), each with sides numbered 1 through 6. All four dice are rolled independently. Let S be the set of the three smallest values obtained from these four rolls. Calculate the exact probability that the sum of the elements in S equals 3.

Requirements

  1. Provide the exact probability as an irreducible fraction.
  2. Detail the logic used to identify the valid combinations of dice outcomes.
  3. Address how your calculation changes if the dice were not fair (e.g., if the probability of rolling a '1' was 1/3 instead of 1/6).

Example Case

If the rolls are (1, 1, 1, 6), the three smallest values are (1, 1, 1). Their sum is 3. This is a valid outcome. If the rolls are (1, 1, 2, 2), the three smallest values are (1, 1, 2). Their sum is 4. This is an invalid outcome.

Edge Cases to Consider

  • Multiple dice showing the same value.
  • The scenario where the largest die value is also 1.

Hints

Hint 1

Identify which combinations of four values result in the three smallest summing to exactly 3.

Hint 2

Consider the ordered multiset of outcomes and use the multinomial coefficient or systematic counting to determine the number of favorable permutations out of the total 6^4 outcomes.

Hint 3

Note that for the sum of the three smallest dice to be 3, the three smallest values must be (1, 1, 1), regardless of the value of the fourth die.

Follow-up questions

  • How does the probability change if the dice are ten-sided (1-10) instead of six-sided?
  • If we roll N dice instead of 4, what is the general expression for the probability that the k smallest dice sum to a specific value T?
  • How would you simulate this problem using a Monte Carlo method to verify your theoretical result, and what is the expected convergence rate?