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# Conditional probability

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## Conditional probability

If we need to find the probability of an event occurring given that another event has already occurred, then we are dealing with **conditional probability.**

If A and B are two events, then the conditional probability that A occurs given that B already has is written as P(A | B) where:

or:

**Example:**

A frog climbing out of a well is affected by the weather. When it rains, he falls back down the well with a probability of 1/10. In dry weather, he only falls back down with probability of 1/25. The probability of rain is 1/5.

**Find the probability that given he falls it was a rainy day.**

Let's start by drawing the tree diagram of these events:

**Another way to think about conditional probability is:**

How many ways there are of obtaining 'A and B' out of a total possible number of ways of getting the 'B':

**If we rearrange this formula we obtain another useful result:**

**If the two events A and B are independent (i.e. one doesn't affect the other) then quite clearly:**