Conditional Probability

Topics: Probability

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If we add additional information to an experiment, we change the probability of the event.

(definition)

Let $A,B$ be events such that $\mathbb{P}(B)>0$.

We define the conditional probability of the event $A$, given (i.e. knowing, supposing, assuming) the occurrence of the event $B$, as follows:

$$\mathbb{P}(A\mid B):=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}$$

$\mathbb{P}(A\mid B)$ is read “the probability of $A$ given $B$“.

Example

Let’s throw a die twice.

What’s the probability of obtaining a $6$ in one of the throws, given different numbers in both throws? This is a conditional probability.

Let:

• $A:$ get a $6$
• $B:$ get different numbers in both throws

First, note that $B=\{(x,y)\mid x,y\in \{1,\dots ,6\},x\ne y\}$ ($|B|=30$). Thus:

$$\mathbb{P}(B)=\frac{|B|}{|\Omega |}=\frac{30}{36}$$

On the other hand, $A=\{(i,6),(6,j),(6,6)\mid i,j\in \{1,\dots ,5\}\}$. Thus:

$$A\cap B=\{(i,6),(6,j)\mid i,j\in \{1,\dots ,5\}\}$$

$$⟹\mathbb{P}(A\cap B)=\frac{|A\cap B|}{|\Omega |}=\frac{10}{36}$$

Finally, with the formula for a conditional probability, we have that:

$$\mathbb{P}(A\mid B)=\frac{\frac{10}{36}}{\frac{30}{36}}=\frac{10}{30}=\frac{1}{3}$$

(theorem)

Conditional probability is a probability measure.