Given the probability that A can solve a problem is $2 / 3$, and the probability that B can solve the same problem is \%, find the probability that(i)at least one of $A$ and $B$ will solve the problem(ii)none of the two will solve the problem

Home » Given the probability that A can solve a problem is , and the probability that B can solve the same problem is \%, find the probability that(i)at least one of and will solve the problem(ii)none of the two will solve the problem

Given the probability that A can solve a problem is $2 / 3$ , and the probability that B can solve the same problem is \%, find the probability that
(i)at least one of $A$ and $B$ will solve the problem
(ii)none of the two will solve the problem

Given the probability that A can solve a problem is $2 / 3$ , and the probability that B can solve the same problem is \%, find the probability that
(i)at least one of $A$ and $B$ will solve the problem
(ii)none of the two will solve the problem

Given : Here probability of $A$ and $B$ that can solve the same problem is given, i.e., $P(A)=\frac{2}{3}$ and $P(B)=\frac{3}{5} \Rightarrow P($ $\bar{A})=\frac{1}{3}$ and $\mathrm{P}(\bar{B})=\frac{2}{5}$
Also, $A$ and $B$ are independent. not $A$ and not $B$ are independent.
i) atleast one of $A$ and B will solve the problem
Now, P(atleast one of them will solve the problem) $=1$ – P(both are unable to solve)
$\begin{array}{l} =1-\mathrm{P}(\bar{A} \cap \bar{B}) \\ =1-\mathrm{P}(\bar{A}) \times \mathrm{P}(\bar{B}) \\ =1-\left(\frac{1}{3} \times \frac{2}{5}\right) \\ =\frac{13}{15} \end{array}$
Therefore, atleast one of $A$ and $B$ will solve the problem is $\frac{13}{15}$
ii) none of the two will solve the problem
Now, $\mathrm{P}$ (none of the two will solve the problem) $=\mathrm{P}(\bar{A} \cap \bar{B})$
$=\mathrm{P}(\bar{A}) \times \mathrm{P}(\bar{B})$
$\begin{array}{l} =\frac{1}{3} \times \frac{2}{5} \\ =\frac{2}{15} \end{array}$
Therefore, none of the two will solve the problem is $\frac{2}{15}$