Successive amplitudes in quantum mechanics In quantum mechanics we define amplitudes for events, like propagation from one point to some other point. Lets say that from a source to detector we have some amplitude (D/S).
But, lets now say that we have one mid point. Now we can say that the amplitude from S to M (midpoint) and then to D is (D/M)(M/S). Why multiply?
 A: Because amplitudes are related to probabilities, and that's how the laws of probability work.
A fair die has a probability of $1/6$ of landing on any side. If you roll the die twice, the probability of rolling a 6 the first time and rolling a 6 the second time is $1/6\times 1/6=1/36$. Likewise, for a single roll, the probability of rolling a 5 or rolling a 6 is $1/6+1/6=1/3$. So you see that the laws of probability dictate the following:

The probability of one event happening and another mutually-exclusive event happening is the product of the probabilities of the two events happening.
The probability of one event happening or another mutually-exclusive event happening is the sum of the probabilities of the two events happening.

So the probability of the particle traveling from S to M to D is the probability of traveling from S to M and traveling from M to D, hence is equal to the product of the two individual probabilities. Since the squared magnitude of the amplitude is the probability, it should be straightforward to see that amplitudes should follow the same rule, since $|a||b|=|ab|$.
