Question

Probability of solving specific problem independently by $A$ and $B$ are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that the problem is solved.

Answer 1

Let $A$ be the event of solving the problem by $A$ and $B$ be that of $B$.

Then according to the problem,

$P\left(A\right)=\frac{1}{2}$ and $P\left(B\right)=\frac{1}{3}$.

Now if they solve the problem independently then the probability of solving the problem is

$P(A\cup B)$

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

$=P\left(A\right)+P\left(B\right)-P\left(A\right).P\left(B\right)$ [ Since $A$ and $B$ are independent]

$=\frac{1}{2}+\frac{1}{3}-\frac{1}{2}.\frac{1}{3}$

$=\frac{5}{6}-\frac{1}{6}$

$=\frac{4}{6}$

$=\frac{2}{3}$.