Question

If four persons independently solve a certain problem correctly with the probabilities $\frac{1}{2},\frac{3}{4},\frac{1}{4}\mathrm{and}\frac{1}{8}$. 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

The correct option is A

$\frac{235}{256}$

Explanation for the correct answer:

Step 1: Probability of the complement of the Event $A$

Given, $\mathrm{P}\left(\mathrm{A}\right)=\frac{1}{2}$

Hence, $\begin{array}{rcl}\mathrm{P}(\mathrm{A}\text{'})& =& 1-\mathrm{P}\left(\mathrm{A}\right)\end{array}$ where $P(A\text{'})$ complements of the Event $A$

$\begin{array}{rcl}P(A\text{'})& =& 1-\frac{1}{2}\\ & =& \frac{1}{2}\end{array}$

Similarly, $\mathrm{P}\left(\mathrm{B}\right)=\frac{3}{4}$

$$\begin{gathered} \mathrm{P}\text{'}\left(\mathrm{B}\right)=1-\frac{3}{4} \\ \mathrm{P}\text{'}\left(\mathrm{B}\right)=\frac{1}{4} \end{gathered}$$

Similarly, $\mathrm{P}\left(\mathrm{C}\right)=\frac{1}{4}$

$$\begin{gathered} \mathrm{P}\text{'}\left(\mathrm{C}\right)=1-\frac{1}{4} \\ \mathrm{P}\text{'}\left(\mathrm{C}\right)=\frac{3}{4} \end{gathered}$$

Similarly, $\mathrm{P}\left(\mathrm{D}\right)=\frac{1}{8}$

$$\begin{gathered} \mathrm{P}\text{'}\left(\mathrm{D}\right)=1-\frac{1}{8} \\ \mathrm{P}\text{'}\left(\mathrm{D}\right)=\frac{7}{8} \end{gathered}$$

Step 2: Finding the required probability

Probability(at least one of them solves correctly) $=1-$ Probability(None of them solves correctly)

P(at least one of them solves correctly)

$$\begin{gathered} =1-\left(\frac{1}{2}\times \frac{1}{4}\times \frac{3}{4}\times \frac{7}{8}\right) \\ =1-\left(\frac{21}{256}\right) \\ =\frac{256-21}{256} \\ =\frac{235}{256} \end{gathered}$$

Hence, the correct answer is an option (A).