Four persons independently solve a certain problem correctly with probabilities \frac{1}{2} , \textrm{ } \frac{3}{4} , \textrm{ } \frac{1}{4} , \textrm{ } \frac{1}{8}. Then the probability that the problem is solved correctly by atleast one of them is

Engineering

Mathematics

Addition Theorem on Probability

Question

Four persons independently solve a certain problem correctly with probabilities $\frac{1}{2}, \frac{3}{4}, \frac{1}{4}, \frac{1}{8}$ . Then the probability that the problem is solved correctly by atleast one of them is

$\frac{235}{256}$

$\frac{3}{256}$

$\frac{253}{256}$

$\frac{21}{256}$

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

$P (C) = \frac{1}{4}, P (s) = \frac{1}{8}$

Now, P (problems is solved correctly by atleast one of them)

= 1 – P (none of them solved the problem correctly)

$= 1 - P (\bar{A}) P (\bar{B}) P (\bar{C}) P (\bar{D}) = 1 - (1 - \frac{1}{2}) (1 - \frac{3}{4}) (1 - \frac{1}{4}) (1 - \frac{1}{8})$

$= 1 - (\frac{1}{2} \times \frac{1}{4} \times \frac{3}{4} \times \frac{7}{8}) = 1 - \frac{21}{256} = \frac{235}{256}$