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 .
A product of 2 could happen in 5 different ways, so the probability is .
Same probability applies to an outcome of 3 and 5.
The largest out come is five 6s, which has a probability of
Hope this helps.
Outcome Ways
1 1
2 5
3 5
4 5 + 10
5 5
6 5 + 10 + 10
...
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
Maybe you can take it from there...