Conditional Probability

The outcome of one event depends on an earlier event.

For example, the probability of obtaining a positive test result is dependent on whether or not you have a particular condition. If you have a condition, it is more likely that a test result is positive. We can formulate conditional probabilities for any two events in the following way:

Example

We run a test to see if the person actually has cancer. In blue is the test and the conditionnal probability.

So the test depends on something else: if the person has cancer or not.

If this, we can write the truth table:

To calculate the first Prob for example = P(cancer) = 0.1 * P(positive | Cancer) = 0.9 = 0.09.

The sum of all of those P's = 1.

To find the Prob of a result being positive = 0.09 + 0.18 = 0.27. In reality what we did was:

Ex: say we have a bag of coins with 1 fair coin and 1 non fair coin. We pick 1 at random (50-50 chances). What's the prob of picking a head?

Same question now we flip the coin twice.