I have ten slots, and each slot can take the value 1, 2 or 3. The product of the ten slots is then calculated. What is the probability distribution of all possible products for this system?
Some are easy, for example a product of 1 can only be produced by 1x1x1x1x1x1x1x1x1x1=1. But what about, say, 108?
To start with, I'm assuming that 1, 2 and 3 occur equally likely, but eventally I would like to be able to change their likelihoods and still be able to work out the probability distribution of the products.
Thanks in advance!
This is a trinomial distribution with the added complexity of determining the number of $1s, 2s$ and $3s$ for every "possible" outcome.
Example: $P(108)$
$108 = 2^2\cdot 3^3\cdot 1^5$
Using the trinomial equation:
$$P(108) = \frac{10!}{2!\cdot3!\cdot 5!}\cdot (\frac{1}{3})^{10} = .042676$$