Day 1 · Highest Roll EV¶
Edited by Sophia
| Study record | Value |
|---|---|
| Question | 3 of 4 |
| Topic | Probability |
| Difficulty | Medium |
| Attempts | 1 |
| Mastery (Notion notation) | ❚ |
| Latest attempt | October 22, 2026 |
| Source | OpenQuant: Highest Roll EV |
Problem¶
Jim will roll a fair, six-sided die until he gets a 4. What is the expected value of the highest number he rolls through this process?
Solution¶
Since the game stops as soon as a 4 is rolled, 4 is always rolled at least once, meaning the maximum value is at least 4. Any rolls of 1, 2, or 3 have no impact on the maximum. Therefore, the problem simplifies entirely to the relative arrival order of the numbers 4, 5, and 6. By symmetry, all \(3! = 6\) relative orderings of \(\{4, 5, 6\}\) are equally likely:
-
If 4 appears before both 5 and 6 (orders:
4-5-6,4-6-5\(\to 2\) outcomes), neither 5 nor 6 appears before the game ends, so the maximum is 4. -
If 5 appears before 4, but 6 appears after 4 (order:
5-4-6\(\to 1\) outcome), a 5 was rolled but no 6 was rolled, so the maximum is 5. -
If 6 appears before 4 (orders:
6-4-5,6-5-4,5-6-4\(\to 3\) outcomes), a 6 was rolled, so the maximum is 6. Taking the expectation across these 6 outcomes: 31/6.