# What does multiplication mean in probability theory?

For independent events, the probability of both occurring is the product of the probabilities of the individual events:

$Pr(A\; \text{and}\;B) = Pr(A \cap B)= Pr(A)\times Pr(B)$.

Example: if you flip a coin twice, the probability of heads both times is: $1/2 \times 1/2 =1/4.$

I don’t understand why we multiply. I mean, I’ve memorized the operation by now, that we multiply for independent events; but why, I don’t get it.

If I have $4$ bags with $3$ balls, then I have $3\times 4=12$ balls: This I understand. Multiplication is (the act of) scaling.

But what does scaling have to do with independent events? I don’t understand why we scale one event by the other to calculate $Pr(A \cap B)$, if A, B are independent.

Explain it to me as if I’m really dense, because I am. Thanks.

I like this answer taken from http://mathforum.org/library/drmath/view/74065.html :

It may be clearer to you if you think of probability as the fraction
of the time that something will happen. If event A happens 1/2 of the
time, and event B happens 1/3 of the time, and events A and B are
independent, then event B will happen 1/3 of the times that event A
happens, right? And to find 1/3 of 1/2, we multiply. The probability
that events A and B both happen is 1/6.

Note also that adding two probabilities will give a larger number than
either of them; but the probability that two events BOTH happen can’t
be greater than either of the individual events. So it would make no
sense to add probabilities in this situation.