Question

State and prove Addition theorem on probability.

Answer 1

Addition theorem of probability $\to$ If $A$ and $B$ are any two events then the probability of happening of at least one of the events is defined as

$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

Proof:-

From set theory, we know that,

$n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$

Dividing the above equation by $n\left(S\right)$ both sides we have

$\frac{n(A\cup B)}{n\left(S\right)}=\frac{n\left(A\right)}{n\left(S\right)}+\frac{n\left(B\right)}{n\left(S\right)}-\frac{n(A\cap B)}{n\left(S\right)}$

$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)(\because P\left(X\right)=\frac{n\left(X\right)}{n\left(S\right)})$

Hence proved.