Let a fair die be rolled 5 times. Find the math expectation of the product of the outcomes.
If the outcomes are 1 1 1 1 1 then the product is 1 with a probability of $\displaystyle \frac{1}{6^5}$.
A product of 2 could happen in 5 different ways, so the probability is $\displaystyle \frac{5}{6^5}$.
Same probability applies to an outcome of 3 and 5.
The largest out come is five 6s, which has a probability of $\displaystyle \frac{1}{6^5}$
Hope this helps.
Outcome Ways
1 1
2 5
3 5
4 5 + 10
5 5
6 5 + 10 + 10
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Hi Aryth,
Let's see if we can solve the corresponding problem for 2 dice. With luck, maybe this will show us the way to a more general solution.
To that end, let's say x and y are the numbers on the two dice. We assume each pair (x,y) has equal probability, i.e. 1/36. By definition, the expected value of their product is
$\displaystyle E(XY) = \sum_{x,y} xy \; p(x,y)$
$\displaystyle = \sum_{x=1}^6 \sum_{y=1}^6 \frac{xy}{36}$
$\displaystyle = \frac{1}{36} \left( \sum_{x=1}^6 x \right) \left( \sum_{y=1}^6 y \right)$
$\displaystyle = \frac{1}{36} \left( \frac{6 \cdot 7}{2} \right) ^2$
Maybe you can take it from there...