Probability within a series of events

Conditional probability allows us to find the probability of an event in within a sub-event of a larger sample space, if we know the sub-event already occurred.

For instance, if we say two children are born and one of them is a boy, what are the chances that the other is a girl, assuming each is equally probable?

Here is all the possible combinations of boy girl we could have:

$${ BB, BG, GB, GG}$$ So our sub-event in this case is that one of them is a boy. That narrows our sample space down to: $${BB,BG,GB}$$ So what is the probability that a girl is chosen? It looks like the answer is 2/3!!

This might surprise some people, but you have to remember, we didn't specify the order in which the boy was conceived.

Calculating the probability

It's helpful to think of equation for conditional probability to be similar to the inverse of the product rule.

$$ P(A)=\frac{P(A\cap B)}{P(A| B)} $$ Lets break it down: $A\cap B$ - The event that occurs in the sub-event. $A | B$ - The outer event that we know occurs before the inner event.

Bayes' Theorem This theorem adds another layer to the conditional probability equation: What if we don't know what sub event occurred, what if it was only probable?