Question

For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is $\frac{4}{5},$ then the probability that he is unable to solve less than two problems is

• A. $\frac{201}{5}\left(\frac{1}{5}\right)^{49}$
• B. $\frac{164}{25}\left(\frac{1}{5}\right)^{48}$
• C. $\frac{316}{25}\left(\frac{4}{5}\right)^{48}$
• D. $\frac{54}{5}\left(\frac{4}{5}\right)^{49}$

Answer 1

Answer: (D)

Let, $X=$ The number of questions, student are unable to solve.
$p=P \, $ (the student answers a question correctly) $=\frac{4}{5}$
$\Rightarrow \, q=1-p=\frac{1}{5}$ (probability that the student does not answer a question correctly)
Also, $n=50$ (Total number of question)
Hence, required probability:
$P\left(\right.X < 2\left.\right)$
$ \, =P\left(X = 0\right)+P\left(X = 1\right)$
$={ }^{50} C_{0}\left(\frac{4}{5}\right)^{50}+{ }^{50} C_{1}\left(\frac{4}{5}\right)^{49}\left(\frac{1}{5}\right)$
$=\frac{54}{5}\left(\frac{4}{5}\right)$