Conditional Probability

Conditional probability is used to calculate probability when we have information that can make an event more likely.

For example:

What are the chances of guessing the number rolled on an even 6 sided die?

Since the sample space is {1,2,3,4,5,6} the chances are 1/6

But suppose the person rolling it gave you a hint and told you the number was odd, now what are your chances?

The sample space of odd numbers is {1,3,5} so your chances are 1/3

This is evident in such a small sample space, but we can use a formula for larger sample spaces.

P(E|F) = P(E and F)/P(F)

P(E|F) is read as probability of E given F. In this case E is the chance of guessing the number and F is the chance of the number being odd.

The probability of getting a number that is odd is 1/2 since half the numbers on a six sided die are odd. So P(F)=1/2

The probability of the number being odd and of you guessing the number is still (1/6)
so P(E and F) is 1/6

using the formula, the conditional probability P(E|F) that you can guess the number (E) given that you know it is odd(F) is

P(E|F) = (1/6) / (1/2) = (1/3)

So the formula gives us the same answer we previously saw.