Question

Given the probability that A can solve a problem is ${\displaystyle \frac{2}{3}}$ and the probability that B can solve the same problem is ${\displaystyle \frac{3}{5}}$. Find the probability that none of two will be able to solve the problem.

Answer 1

Hint: Here we solve the problem by finding the individual probability of events for not solving the problem because here the events have solved the problem individually. If suppose $p(x)$is the probability of solving some x work then 1-p(x) will be the probability for not solving work.

Complete step-by-step answer:

Now here let us consider that A be an event that solves problem A and also B be an event that solves problem B.

According to the data we can say that
 Probability that the event A can solve problem A is ${\displaystyle \frac{2}{3}}$ $\Rightarrow P(A)={\displaystyle \frac{2}{3}}$
Probability that the event B can solve the problem B is${\displaystyle \frac{3}{5}}$ $\Rightarrow P(B)={\displaystyle \frac{3}{5}}$

Here we have to find the probability that none of two will be able to solve the problem.

So here we have to find the value of $P(\overline{A}.\overline{B})$.
$$\Rightarrow P(\overline{A}.\overline{B})=P(\overline{A})P(\overline{B})$$

Since we know that the event A and B has solved the problem individually so we should also find their individual probability of not solving the problem.

$$\Rightarrow P(\overline{A}.\overline{B})=P(\overline{A})P(\overline{B})$$
$\Rightarrow P(\overline{A}.\overline{B})=(1-P(A))(1-P(B))$
$\Rightarrow P(\overline{A}.\overline{B})=\left(1-{\displaystyle \frac{2}{3}}\right)\left(1-{\displaystyle \frac{3}{5}}\right)$
$\Rightarrow P(\overline{A}\overline{B})={\displaystyle \frac{1}{3}}\times {\displaystyle \frac{2}{5}}$
$\Rightarrow P(\overline{A}\overline{B})={\displaystyle \frac{2}{15}}$

Therefore the probability that none of the two events A and B will be able to solve the problem = ${\displaystyle \frac{2}{15}}$.

Note: In this problem there are two events A and First we have to observe that event A and event B are working independently, which means the probability of event A solving (or not solving) the problem is entirely independent of the probability of event B solving (or not solving) the problem. Based on this we have to find the values and use them according to it.