Q69 / 155 · WBJEE 2019 · Mathematics
A problem in mathematics is given to 4 students whose chances of solving individually are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and $\frac{1}{5},$ The probability that the problem will be solved at least by one student is
- A$\frac{2}{3}$
- B$\frac{3}{5}$
- Correct answerC$\frac{4}{5}$
- D$\frac{3}{4}$
Solution
$$
\begin{aligned}
&\text { (c) Let } P(A)=\frac{1}{2}, P(B)=\frac{1}{3}\\
&\begin{aligned}
P(C) &=\frac{1}{4}, P(D)=\frac{1}{5} \\
\text { Now, } P(A&\cup B \cup C \cup D) \\
&=1-P(\bar{A} \cap \bar{B} \cap \bar{C} \cap \bar{D}) \\
&=1-P(\bar{A}) P(\bar{B}) P(\bar{C}) P(\bar{D}) \\
&=1-\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right) \\
&=1-\left(\frac{1}{2}\right)\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\left(\frac{4}{5}\right)=1-\frac{1}{5}=\frac{4}{5}
\end{aligned}
\end{aligned}
$$
Hence, the correct answer is option: 'C'.