Conditional probability

The conditional probability of [math]B [/math] given [math]A [/math] is the probability of [math]B ∩ A [/math], "scaled up" so that the probability of [math]A [/math] given [math]A [/math] is 1.

Formally:

Definition: Conditional probability

If and are events, then the conditional probability of given (written ) is given by

For example, suppose we wish to model the following experiment: we first select one of two coins. The first coin (coin a) is weighted: it lands heads 3/4 of the time. The second coin (coin b) is fair: it lands heads 1/2 of the time. We choose the first coin 1/3 of the time. We want to find the probability of getting heads.

How do we interpret the facts given in the problem?

We first construct a sample space: there are 4 things that can happen: we can choose coin a and flip heads, we can choose coin a and tails, we can choose coin b and flip heads, or we could choose coin b and flip tails. A reasonable sample space would be .

It is (always) helpful to define some events: let be the event that we pick coin a, and be the event that we flip heads; define [math]B [/math] and [math]T [/math] similarly.

Now we need to interpret the probabilities given in the problem. When we say "[coin a] lands heads 3/4 of the time", we don't mean that 3/4 of the time we choose coin a and flip it and get heads (this would be ). Rather, we mean that if we restrict our attention to the outcomes where we chose coin a, then the probability of getting heads in that restricted experiment is 3/4. Put more simply, the probability that we get heads given that we choose coin a is 3/4.

We interpret this in our model by setting . Since we choose coin a with probability 1/3, we see that : we would expect to select coin a and flip heads in about a quarter of the experiments.

Similarly, so .

Since we can only select one of the coins, the events [math]A [/math] and [math]B [/math] are disjoint, so we can use the third Kolmogorov axiom to compute :