Question

Four persons independently solve a certain problem correctly with probabilities $\dfrac{1}{2},\dfrac{3}{4},\dfrac{1}{4},\dfrac{1}{8}$. Then the probability that the problem is solved correctly by at least one of them is
(A). $\dfrac{{235}}{{256}}$
(B). $\dfrac{{21}}{{256}}$
(C). $\dfrac{3}{{256}}$
(D). $\dfrac{{253}}{{256}}$

Answer 1

Hint: Before attempting this question, one should have prior knowledge about the concept of the probability and also remember to use the rule of complementary events i.e. $P\left( {A'} \right) + P\left( A \right) = 1$, use this information to approach the solution of the problem.

Complete step-by-step answer:
According to the given information we know that four persons solves problem independently with probabilities $\dfrac{1}{2},\dfrac{3}{4},\dfrac{1}{4},\dfrac{1}{8}$
So, let P (A) = $\dfrac{1}{2}$, P (B) = $\dfrac{3}{4}$, P (C) = $\dfrac{1}{4}$and P (D) = $\dfrac{1}{8}$
As we know by the rule of complementary events $P\left( {A'} \right) + P\left( A \right) = 1$
We know that probability of getting wrong solution of the problem = 1 – probability of person solving the problem
Therefore, probability of getting wrong solution by person a i.e. P (A’) = $1 - \dfrac{1}{2}$
Probability of getting wrong solution by person a i.e. P (A’) = $\dfrac{1}{2}$
Probability of getting wrong solution by person b i.e. P (B’) = $1 - \dfrac{3}{4}$
So, Probability of getting wrong solution by person b i.e. P (B’) = $\dfrac{1}{4}$
Probability of getting wrong solution by person c i.e. P (C’) = $1 - \dfrac{1}{4}$
So, Probability of getting wrong solution by person c i.e. P (C’) = $\dfrac{3}{4}$
Probability of getting wrong solution by person d i.e. P (D’) = $1 - \dfrac{1}{8}$
So, Probability of getting wrong solution by person d i.e. P (D’) = $\dfrac{7}{8}$
Probability of that the none person can solve the problem = P (A’) $ \times $ P (B') $ \times $ P (C’) $ \times $ P (D’)
Substituting the value in the above equation we get
Probability of that the none person can solve the problem = \[\dfrac{1}{2} \times \dfrac{1}{4} \times \dfrac{3}{4} \times \dfrac{7}{8}\]
So, Probability of that the none person can solve the problem = \[\dfrac{{21}}{{256}}\]
The probability of that at least one of the persons will be able to solve the problem = 1 – Probability of that the none person can solve the problem
Therefore, Probability of that at least one of the persons will be able to solve the problem = $1 - \dfrac{{21}}{{256}}$
Probability of that at least one of the persons will be able to solve the problem = $\dfrac{{235}}{{256}}$
Hence, option A is the correct option.

Note: In the above solution we used the basic logic reasoning to find the probability to find the required probability and as we know that probability of getting the correct and wrong answer is equal to 1 that is named as the rule of complementary events. So, to find the probability that at least one of the persons will be able to solve the problem we found the probability of that the none person can solve the problem then we used the rule of complementary events i.e. $P\left( {A'} \right) + P\left( A \right) = 1$ where is probability of happening of event and the probability of event not happening.