Expected Payoff of a Three N-Sided Dice Matching Game
Company: Sig
Role: Quantitative Trader
Category: Statistics & Math
Difficulty: medium
Interview Round: Online Assessment
You roll three fair $N$-sided dice, with faces numbered 1 through $N$. If all three dice show the same number, you receive $A$ dollars. If exactly two of them show the same number, you receive $B$ dollars. If all three numbers are different, you receive $C$ dollars; in another version of the question you pay $C$ dollars instead. What is the expected payoff of a single roll?
Give the answer as an expression in $N$, $A$, $B$ and $C$ for both versions, then evaluate it for the values you are given. The specific numbers are generated randomly, and each question states the required rounding.
```hint Classify the ordered outcomes
Count the equally likely ordered outcomes that fall into each of the three categories, and confirm that the three counts add up to the total.
```
### Constraints and Clarifications
- $N$ is a positive integer, and the three dice are fair and independent.
- $A$, $B$ and $C$ are nonnegative amounts. The version of the question decides whether the all-different outcome pays $C$ or costs $C$.
- There is a single roll and no separate entry fee unless the question states one.
### Clarifying Questions
- Does "exactly two the same" exclude the case where all three match?
- In the version where the all-different outcome costs money, is $C$ the whole loss, or is an entry fee charged as well?
- Should the answer be rounded only at the end, and to how many decimal places?
### What a Strong Answer Covers
- Correct probabilities for the three outcome classes for a general $N$, with a check that they sum to one
- The expected value as a probability-weighted sum, with the sign of the all-different term right in both versions
- Simplification to a single expression over a common denominator
- Small values of $N$ for which one of the classes cannot occur
### Follow-up Questions
- What entry fee would make the game fair?
- What is the variance of the payoff?
- With four dice, what is the probability that exactly two show the same number and the other two differ from it and from each other?
Overview: An expected value question about rolling three fair N-sided dice, with different payoffs when all three match, exactly two match, or all differ. It tests counting the outcome classes correctly for a general number of faces and combining them into one expected payoff, including a variant where the all-different outcome costs money.
Read the full Sig Quantitative Trader interview experience this question came from