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Question: Given the probability that A can solve a problem is \(\dfrac{2}{3}\) and the probability that B can ...

Given the probability that A can solve a problem is 23\dfrac{2}{3} and the probability that B can solve the same problem is 35\dfrac{3}{5}. Find the probability that none of two will be able to solve the problem.

Explanation

Solution

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)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 23\dfrac{2}{3} ⇒P(A)=23 \Rightarrow P(A) = \dfrac{2}{3}
Probability that the event B can solve the problem B is35\dfrac{3}{5} ⇒P(B)=35 \Rightarrow P(B) = \dfrac{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(Aˉ.Bˉ)P(\bar A.\bar B).
⇒P(Aˉ.Bˉ)=P(Aˉ)P(Bˉ)\Rightarrow P(\bar A.\bar B) = P(\bar A)P(\bar 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.

⇒P(Aˉ.Bˉ)=P(Aˉ)P(Bˉ) \Rightarrow P(\bar A.\bar B) = P(\bar A)P(\bar B)
⇒P(Aˉ.Bˉ)=(1−P(A))(1−P(B))\Rightarrow P(\bar A.\bar B) = (1 - P(A))(1 - P(B))
⇒P(Aˉ.Bˉ)=(1−23)(1−35)\Rightarrow P(\bar A.\bar B) = \left( {1 - \dfrac{2}{3}} \right)\left( {1 - \dfrac{3}{5}} \right)
⇒P(AˉBˉ)=13×25\Rightarrow P(\bar A\bar B) = \dfrac{1}{3} \times \dfrac{2}{5}
⇒P(AˉBˉ)=215\Rightarrow P(\bar A\bar B) = \dfrac{2}{{15}}

Therefore the probability that none of the two events A and B will be able to solve the problem = 215\dfrac{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.