Law of total probability

Let's try to generalise this example a bit. We will call left-handedness "condition A", and gender is condition B. For condition A, we are only interested in one state, but condition $B$ has two states (for the sake of this exercize) $B_1$ (male) and $B_2$(female).

Probability of A and $B_1$, as we showed above, is thus:

And then the reverse:

We need to add these two together to get the complete $P(A)$.

Now, let's imagine a different scenario where we have 3 mutually exclusive conditions $B_1,B_2,B_3$.

In order to get the total $P(A)$ we need to add them all together:

If we generalize this as a sum with number of conditions $i$:

We call this the Law of Total Probability.