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(A) Let $P(A), P(B)$,and $P(C)$ be the probabilities of students $A, B$,and $C$ solving the problem respectively.$P(A) = \frac{1}{2}, P(B) = \frac{1}{

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$\frac{3}{4}$

Vedclass · Answer

(A) Let $P(A), P(B)$,and $P(C)$ be the probabilities of students $A, B$,and $C$ solving the problem respectively.$P(A) = \frac{1}{2}, P(B) = \frac{1}{3}, P(C) = \frac{1}{4}$.The problem is solved if at least one of them solves it.It is easier to calculate the probability that the problem is not solved by anyone.The probability that $A$ does not solve the problem is $P(A') = 1 - \frac{1}{2} = \frac{1}{2}$.The probability that $B$ does not solve the problem is $P(B') = 1 - \frac{1}{3} = \frac{2}{3}$.The probability that $C$ does not solve the problem is $P(C') = 1 - \frac{1}{4} = \frac{3}{4}$.Since these events are independent,the probability that none of them solve the problem is $P(	ext{None}) = P(A') 	imes P(B') 	imes P(C') = \frac{1}{2} 	imes \frac{2}{3} 	imes \frac{3}{4} = \frac{6}{24} = \frac{1}{4}$.The probability that the problem is solved is $P(	ext{Solved}) = 1 - P(	ext{None}) = 1 - \frac{1}{4} = \frac{3}{4}$.

$A$ problem in Mathematics is given to three students $A, B$,and $C$. Their respective probabilities of solving the problem are $\frac{1}{2}, \frac{1}{3}$,and $\frac{1}{4}$. What is the probability that the problem is solved?

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