Conditional Probability

Conditional probability is the probability of something, given that you already know something else. The bar is read as given. Suppose a student is chosen from these fifty, and you are told the student is in Year 10. Now only the Year 10 row matters. It becomes the new whole, twenty five students. Twelve of them play an instrument, so the probability of an instrument, given Year 10, is twelve out of twenty five. Knowing the condition shrinks the sample space.

Turn the question around. Given that the student plays an instrument, what is the probability the student is in Year 10? Now the instrument column is the new whole: twenty students. Twelve of them are in Year 10, so the answer is twelve twentieths, or three fifths. The top, twelve, is the same as before, but the bottom is different, so the answers are different. The event after the bar decides the new whole.

On a Venn diagram, given B means you only look inside circle B. That circle holds six in the overlap and six in B only, twelve people in all. The ones who are also in A are the six in the overlap, so the probability of A given B is six twelfths, or one half. For B given A, look only inside circle A: nine plus six is fifteen, and six of them are in B, so the probability is six fifteenths, or two fifths. Again the top is the same and the bottom changes.

When you are given probabilities instead of counts, use the formula: the probability of A given B equals the probability of A and B, divided by the probability of B. On the Venn diagram, that is six thirtieths divided by twelve thirtieths. The thirtieths cancel, leaving six twelfths, exactly the answer from shrinking the sample space. With decimals, if the probability of A and B is zero point one and the probability of B is zero point four, then the probability of A given B is zero point one divided by zero point four, which is zero point two five.

The train is late with probability zero point two. If it is late, Jo is late for work with probability zero point five. If it is on time, Jo is late with probability zero point one. Those second branches are already conditional: zero point five is the probability Jo is late, given the train is late. Now reverse it. Given that Jo is late, what is the probability the train was late? Jo is late on two paths: zero point one and zero point zero eight, which add to zero point one eight. That is the new whole. The train-late path is zero point one, so the answer is zero point one divided by zero point one eight, which is ten eighteenths, or five ninths. Quiz and exercises now.