Question

What is an example of conditional probability?

Answer 1

Conditional probability:

• The conditional probability of an event is based on a condition.
• The conditional probability of an event $A$ given that $B$ has occurred is given by $\mathbf{\text{P}}\left(A|B\right)\mathbf{=}\frac{\mathbf{P}\left(A\cap B\right)}{\mathbf{P}\left(B\right)}$

An example of conditional probability:

One dice is rolled, then the probability of a getting $2$ given that the number is even.

Let's consider,

$A$: rolled number is $2$

$B$: The number is even

Then, $A=\left\{2\right\},B=\{2,4,6\}$

$\Rightarrow$ $A\cap B=\left\{2\right\}$ ,

$P\left(A\cap B\right)=\frac{\mathrm{Favorable}\mathrm{outcomes}}{\mathrm{Total}\mathrm{outcomes}}=\frac{1}{6}$ ,

$P\left(B\right)=\frac{\mathrm{Favorable}\mathrm{outcomes}}{\mathrm{Total}\mathrm{outcomes}}=\frac{3}{6}$

Therefore,

$\mathbf{P}\left(A|B\right)\mathbf{=}\frac{\mathbf{P}\left(A\cap B\right)}{\mathbf{P}\left(B\right)}$

$\Rightarrow$ $P\left(A|B\right)=\frac{{\displaystyle \frac{1}{6}}}{{\displaystyle \frac{3}{6}}}$

$\Rightarrow$ $P\left(A|B\right)=\frac{1}{3}$