Conditional probability in Venn diagrams

In a nutshell

Conditional probabilities can be found using Venn diagrams. In order to do this, you can find the required probability by looking at a restricted part of the sample space.

Procedure

1.	Draw a Venn Diagram with the data you have.
2.	Determine the probability of the intersection of the two events:
3.	Divide that value by the probability of the first event (B):

Example

Based on the following Venn diagram, what is the value of $$P(B|A)$$?

This means find the probability of $$B$$ occurring given that $$A$$ has already happened. First find $$P(A)$$, and then of these the probability of $$B$$ occurring would be represented by $$P(A\cap B)$$.

$$P(A) = 0.4, P(A\cap B)=0.1$$

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Divide the two values:

$$P(B|A)=\dfrac{0.1}{0.4}=0.25$$

The probability of $$B$$ happening given that $$A$$ has already happened is $$\underline {P(B|A)={0.25}}$$.