Calculate the Probability That Two Dice Sum to Seven
Company: Optiver
Role: Software Engineer
Category: Statistics & Math
Difficulty: hard
Interview Round: Technical Screen
Roll two independent fair six-sided dice. What is the probability that their sum is `7`?
Show the sample space and favorable outcomes. Then explain why adding the individual face probabilities would be incorrect.
### Constraints & Assumptions
- The dice are distinguishable.
- Each ordered pair of faces is equally likely.
- Faces are numbered `1` through `6`.
```hint Decide what one elementary outcome is
State whether `(2,5)` and `(5,2)` are the same or different before counting.
```
### What a Strong Answer Covers
- A correctly defined equally likely sample space.
- A complete set of disjoint favorable outcomes.
- The reduced probability and a clear independence argument.
### Follow-up Questions
- Which sum is most likely, and why?
- What is the probability of a sum at least seven?
- How does the answer change if the dice have different numbers of faces?
Quick Answer: Roll two independent fair six-sided dice. Explain the assumptions and derivation clearly, check edge cases, and show how the result changes when those assumptions no longer hold.