Simply put, Conditional Probability is the probability of some event B happens, given some event A has already occurred.

Notation: “the probability of some event B, given some event A has occurred” will be written as P(B|A). Notice that this is a vertical line and not a line meant to represent divide by. We can apply the equations from Rule 6 above to solve probabilities.

Example 1: Let us say that the probability of getting an A and being female is 0.20 and the probability of being female is 0.6. What is the probability of getting an A, given that a student is female?

\(P(A|F)=\frac {P(A\ and\ F)}{P(F)}=\frac{0.2}{0.6}=0.33\)

The concept of conditional probability provides us with the precise definition of independence: Two events A and B are independent if P(B|A) = P(B).

Example 2: You come home late one night from being out (I presume you were studying, but your call!) and you are struggling to find the correct key to your apartment. You have 5 keys. What is the probability that you find the correct key on your second try, assuming you don't repeat keys?

Let A 1 = first key works and A 2 = second key works. The P(A 1 ) = 1/5 and P(A 2 ) = \(P(A2)=P(A_2|A_1^c)=1/4\). This is correct since after you try the first key and it does not work you are left with 4 keys to try. Now to find the probability that you try exactly two keys is the "probability that the first key does not work and the second key does work." Or by probability expressions this is written:

\(P(A_1^c\ and\ A_2)=P(A_1^c) \times P(A_2|A_1^c)=(4/5)\times(1/4)=1/5\)

Surprisingly, the probability of using exactly 2 keys = probability of exactly 3 keys = probability of exactly 4 keys = probability of exactly 5 keys! Bet your friends on that!