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 at least one of them is

• A. $\frac{235}{256}$
• B. $\frac{21}{256}$
• C. $\frac{3}{256}$
• D. $\frac{253}{256}$

Answer 1

Answer: (A)

$\frac{235}{256}$

$P\left(A\right)=1/2\Rightarrow P\left(\overline{A}\right)=1/2P\left(B\right)=3/4\Rightarrow P\left(\overline{B}\right)=1/4P\left(C\right)=1/4\Rightarrow P\left(\overline{C}\right)=3/4P\left(D\right)=1/8\Rightarrow P\left(\overline{D}\right)=7/8$
$P$(at least one of them solves correctly) $=1-P$(none of them solves correctly)
$=1-(\frac{1}{2}\times \frac{1}{4}\times \frac{3}{4}\times \frac{7}{8})=\frac{235}{256}$