Probability 3 Question 8
8. 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
(a) $\frac{235}{256}$
(b) $\frac{21}{256}$
(c) $\frac{3}{256}$
(d) $\frac{253}{256}$
(2013 Adv)
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Solution:
- PLAN It is simple application of independent event, to solve a certain problem or any type of compitition each event in independent of other.
Formula used
$P(A \cap B)=P(A) \cdot P(B)$, when $A$ and $B$ are independent events.
Probability that the problem is solved correctly by atleast one of them $=1-$ (problem is not solved by all)
$\therefore \quad P($ problem is solved $)=1-P($ problem is not solved $)$
$$ \begin{aligned} & =1-P(\bar{A}) \cdot P(\bar{B}) \cdot P(\bar{C}) \cdot P(\bar{D}) \\ & =1-\frac{1}{2} \cdot \frac{1}{4} \cdot \frac{3}{4} \cdot \frac{7}{8}=1-\frac{21}{256}=\frac{235}{256} \end{aligned} $$
