Question

How do you find the probability of three events $?$

Answer 1

Probability of three events:

If $A$,$B$ and $C$ are the three events, then the probability of three events can be calculated as:

$\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cup }\mathit{B}\mathbf{\cup }\mathit{C}\mathbf{)}\mathbf{=}\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{+}\mathit{P}\mathbf{\left(}\mathit{B}\mathbf{\right)}\mathbf{+}\mathit{P}\mathbf{\left(}\mathit{C}\mathbf{\right)}\mathbf{-}\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{)}\mathbf{-}\mathit{P}\mathbf{(}\mathit{B}\mathbf{\cap }\mathit{C}\mathbf{)}\mathbf{-}\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{C}\mathbf{)}\mathbf{+}\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{\cap }\mathit{C}\mathbf{)}$.

If $A$,$B$ and $C$ are the three independent events, then the probability of the event is calculated by:

$\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{\cap }\mathit{C}\mathbf{)}\mathbf{=}\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathit{P}\mathbf{\left(}\mathit{B}\mathbf{\right)}\mathit{P}\mathbf{\left(}\mathit{C}\mathbf{\right)}$

For Example; If $A,B$and $C$ denote the event of getting an even number, odd number, prime number on rolling a dice respectively.

$P\left(A\right)=\frac{3}{6}=\frac{1}{2},P\left(B\right)=\frac{3}{6}=\frac{1}{2},P\left(C\right)=\frac{3}{6}=\frac{1}{2},P(A\cap B)=0,P(A\cap C)=\frac{1}{6},P(B\cap C)=\frac{2}{6}=\frac{1}{3},P(A\cap B\cap C)=0$

$\begin{array}{rcl}P(A\cup B\cup C)& =& \frac{1}{2}+\frac{1}{2}+\frac{1}{2}-0-\frac{1}{3}-\frac{1}{6}+0\\ & =& 1\end{array}$

Hence, in this way probability of three events can be found out.