Connection between multiplication rule in counting and the multiplication rule for the probability that independent events occur together?

Question

I was told that there is a connection between the multiplication rule in counting and the multiplication rule for calculating the probability that independent events occur together. I'm not too sure what this question means or how to interpret it. Any suggestions?

Answer 1

The connection is evident if you consider the following sample space and probability measure. One of $m$ equally likely events happens from one category, and one of $n$ equally likely events happens from another category. We're interested in the probability that one of $k \leq m$ designated events happened from the first category together with one of $\ell \leq m$ from the second. Now of course, independently, the probabilities are $\frac km$ and $\frac\ell n$. To find the probability that both happen, we note that the total number of events that can happen is $mn$, by the counting multiplication law, and they are all equally likely, and the number of events we are interested in is $k\ell$, again by the counting multiplication law. So the probability is $\frac{k\ell}{mn}$. Turns out we've just multiplied the numerators and denominators of the probabilities, which is the same as multiplying the probabilities.

E.g., you flip a coin and roll a die; what is the probability that you flip heads and roll a multiple of $3$? The number of ways to do this is $1 \cdot 2$, out of a total of $2 \cdot 6$, so the probability is $\frac{1 \cdot 2}{2 \cdot 6} = \frac12\cdot\frac26$