# Expected number of different faces in multiple die roles

I roll 6 6-sided regular dice. What is the expected number of unique up-facing sides I will find among those dice?

## 1 Answer

The easiest way is to calculate the probability you will see one particular number, then use the linearity of expectation to multiply by $6$. What is the chance you see a $1$ with six dice?

• E(seeing a N in six dice rolls) = 6*(1/6) = 1 so your saying E(seeing all numbers) = E(seeing a one) + E(seeing a two)... so E(seeing all numbers) = 6E(seeing a N) = 6 Clear and concise. Thanks! +1 – Double AA Apr 17 '13 at 20:06
• The very good hint was used incorrectly in the comment above. – André Nicolas Apr 17 '13 at 20:12
• @AndréNicolas Please elaborate! – Double AA Apr 17 '13 at 20:14
• @DoubleAA: The chance that you see a $1$ is not $1$. You can certainly have rolls like $222222$ that don't have any $1$'s. The expected number of $1$'s is $1$, but that counts $111111$ as $6$. We want to count it as only yes-1 roll that has a one. So what is the chance you see a $1$ with six dice? – Ross Millikan Apr 17 '13 at 20:18
• @RossMillikan 1- (5/6)^6 no? But aren't I looking for expected value? – Double AA Apr 17 '13 at 20:21