Question

if events A and B are independent such that P(A)=3/5,P(B)=2/3,find P(A union B)​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{A and B are independent events such that}$

$\mathsf{P(A)=\dfrac{3}{5},\;P(B)=\dfrac{2}{3}}$

$\underline{\textbf{To find:}}$

$\mathsf{P(A{\cup}B)}$

$\underline{\textbf{Solution:}}$

$\textsf{Since A and B are independent events,}$

$\mathsf{we have\;P(A{\cap}B)=P(A)\,P(B)}$

$\textsf{By addition theorem of probability,}$

$\mathsf{P(A{\cup}B)=P(A)+P(B)-P(A{\cap}B)}$

$\mathsf{P(A{\cup}B)=P(A)+P(B)-P(A)\,P(B)}$

$\mathsf{P(A{\cup}B)=\dfrac{3}{5}+\dfrac{2}{3}-\dfrac{3}{5}{\times}\dfrac{2}{3}}$

$\mathsf{P(A{\cup}B)=\dfrac{9+10}{15}-\dfrac{2}{5}}$

$\mathsf{P(A{\cup}B)=\dfrac{19}{15}-\dfrac{2}{5}}$

$\mathsf{P(A{\cup}B)=\dfrac{19-6}{15}}$

$\implies\boxed{\mathsf{P(A{\cup}B)=\dfrac{13}{15}}}$