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What does the word 'conditional' mean in plain English?
What could conditional probability mean?
If something is conditional, it means that there is some kind of condition that must have been met for that something to occur.
Similarly, conditional probability is the probability of something happening given the condition that something else has happened.
This sounds a little too abstract - let's have a look at an example to understand it better!
In a group of 50 students, 30 play football and 20 play rugby.
If 15 students play both football and rugby, what is the probability that a randomly chosen student plays rugby given that they play football?
Here we want the conditional probability that a student plays rugby given the condition they play football.
Since we know the selected student plays football (based on the condition), the number of all possible outcomes here is 30 (number of students playing football).
We want the probability of that student also playing rugby, so the student plays both sports.
There are 15 students who play both, so 15 is the number of desired outcomes (the numerator).
The probability that a randomly chosen student plays rugby given that they play football is then 15/30 = 1/2
That wasn't that bad, was it?!
We are essentially just restricting the pool of students out of which we are selecting (based on the condition).
If R denotes 'selected student plays rugby' and F denotes 'selected student plays football', we can write our conditional probability as:
