Why do we multiply probabilities?

Question

Let me take this question:

 The probability of raining tomorrow is 0.2. 
 Also, tomorrow I will toss a fair coin.
 What is the probability that tomorrow it rains and I get a head in the coin toss?

(Let's take that these two events are independent.)

Answer 1

Assume that we have an experiment with $M$ equiprobable outcomes $x_k$, $m$ of them considered favorable, and there is a second experiment with $N$ equiprobable outcomes $y_l$, whereby $n$ of them are considered favorable. The probabilities of success in these two experiments then are ${m\over M}$ and ${n\over N}$, respectively.

Calling these two experiments independent means, by definition, that the $MN$ possible combined outcomes $(x_k,y_l)$ are considered equiprobable. Among these $MN$ combined outcomes there are $mn$ where both experiments turn out successfully. The probability that this happens is $${mn\over MN}={m\over M}\cdot{n\over N}\ .$$

Answer 2

Half of the time that it rains, you'll get tails rather than heads (assuming independence and a fair coin, of course.) So it will only rain and come up heads half as often as it rains. So we multiply the probability of rain by the probability of heads (one half).