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# 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:

![](/files/-Lg3CU13cvlV_tIaA0_u)

Example

![](/files/-Lg35fEwuS4MWVwdpQR6)

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: 

![](/files/-Lg36pu-81uy1mI1ILww)

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: 

![](/files/-Lg3891G4S24nkGmTf8v)

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? 

![](/files/-Lg39gtoq_T182IqdkdB)

Same question now we flip the coin twice. 

![](/files/-Lg3B6C_Cw2862OAm837)