For independent events, the probability of both occurring is the product of the probabilities of the individual events:
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(A∩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.