# What is the probability that the problem is solved?

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A problem in mathematics is given to 3 students whose chances of solving it are $\frac{1}{2},\frac{1}{3},\frac{1}{4},$ What is the probability that the problem is solved?

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Let the respective events of solving the problem be denoted by A, B, C.

Then P(A) = $\frac{1}{2}$ , P(B) = $\frac{1}{3}$ , P(C) = $\frac{1}{4}$

Clearly, A, B, C are independent events and the problem will be considered to have been solved if at least one student solves it.

∴  Required probability = P(A or B or C) = $P(A\,\cup\,B\,\cup\,C)$ = $1- P(\overline{A})P(\overline{B})P(\overline{C})$

$\overline{A}, \overline{B},\overline{C}$ are the respective events of not solving the problem.

Also,

$P(\overline{A}) =1-P(A) =1- \frac{1}{2} =\frac{1}{2}$ ,

$P(\overline{B}) =1-P(B) =1- \frac{1}{3} =\frac{2}{3}$

$P(\overline{C}) =1-P(C) =1- \frac{1}{4} =\frac{3}{4}$

Required probability $1- P(\overline{A})P(\overline{B})P(\overline{C}) =1- \frac{1}{2}\times \frac{2}{3}\times \frac{3}{4} =1-\frac{1}{4}$

$=\;\frac{3}{4}$

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