Question

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

Answer 1

Finding the probability of $3$ independent events:

As we know an independent event does not affect other events

So, to find the probability of independent events we simply multiply their probabilities .

Let $3$ events having probabilities be,

$P\left(A\right),P\left(B\right)\&P\left(C\right)$

Then by multiplying these we will get the probability of three independent events

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

For Example; If three coins are tossed together, what are the probability that the first shows head, the second shows tail and the third shows head?

The probability that the first coin shows head $=\frac{1}{2}$

The probability that the second coin shows tail $=\frac{1}{2}$

The probability that the third coin shows head $=\frac{1}{2}$

$\begin{array}{rcl}\therefore P(A\cap B\cap C)& =& P\left(A\right)\times P\left(B\right)\times P\left(C\right)\\ & =& \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\\ & =& \frac{1}{8}\end{array}$

Hence, the probability of three independent events is $\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{\cap }\mathit{C}\mathbf{)}\mathbf{}\mathbf{=}\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{\times }\mathit{P}\mathbf{\left(}\mathit{B}\mathbf{\right)}\mathbf{\times }\mathit{P}\mathbf{\left(}\mathit{C}\mathbf{\right)}$.