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(AandB)=Pr(AB)=Pr(A)×Pr(B).

Example: if you flip a coin twice, the probability of heads both times is: 1/2×1/2=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 3×4=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(AB), if A, B are independent.

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

Answer

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.

Attribution
Source : Link , Question Author : Emi Matro , Answer Author : blue-sky

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