So far I have seen the following formulas:

Formula 1: 

$P(E \vert F) = \frac{P(EF)}{P(F)}$

Formula 2:

$P(EF)= P(F)P(E \vert F)$

Formula 3: 

$P(E) = P(E \vert F)P(F) + P(E \vert F^c)P(F^c)$

Now in the following example something happens that just doesn't make sense to me nor is it explained:

Suppose that a new policy holder has an accident with in a year of purchasing a policy. What is the probability that they are accident prone?

Previously in part one we found out that the event that a new policy holder has an accident with in a year has a .26 probability.

Let $A\_1$ denote the event that the policy holder will have an accident with in a year of purchasing the policy and let A denote the event that the policy holder is accident prone.

Solution:

$P(A \vert A\_1) = \frac{P(AA\_1)}{P(A\_1)}$ 

The following line is where I loose it and don't understand why A and $A\_1$ flip:

$\frac{P(A)P(A\_1 \vert A)}{P(A\_1)}$

according to formula two it should have been:

$P(AA\_1)= P(A\_1)P(A \vert A\_1)$

What are the scenarios when they flip and WHY?