Question

If A and B are two events of a sample space such that $P\left(A\cup B\right)=\frac{5}{6}$,
$P\left(A\cap B\right)=\frac{1}{3},P\left(B\right)=\frac{1}{3}.\hspace{0.17em}\mathrm{Find}P\left(A\right)\mathrm{and}P\left(A\text{'}\right).$

Answer 1

Given:
P(A $\cup$ B) = $\frac{5}{6}$, P(A $\cap$ B) = $\frac{1}{3}$ and P(B) = $\frac{1}{3}$
On applying the addition theorem of probability, we get:
P(A) = P(A $\cap$ B) + P(A $\cup$ B) $-$ P(B)
= $\frac{1}{3}+\frac{5}{6}-\frac{1}{3}$ = $\frac{5}{6}$

Also,
P$(A\text{'})$ = 1 $-$ P(A)
= $1-\frac{5}{6}=\frac{1}{6}$