Question

It is observed that $25\%$ of the cases related to child labour reported to the police station are solved. If $6$ new cases are reported, then the probability that atleast $5$ of them will be solved is

• A. $\left(\frac{1}{4}\right)^{6}$
• B. $\frac{19}{1024}$
• C. $\frac{19}{2048}$
• D. $\frac{19}{4096}$

Answer 1

Answer: (D)

Here, $n=6$
$p=25 \%=\frac{1}{4}, q=1-p=3 / 4$
Let $X$ be the random variable, which denotes the number of cases that police station are solved.
$\therefore $ According to question,
$P(X=5)+P(X=6)$
$={ }^{6} C_{5}(p)^{5}(q)^{6-5}+{ }^{6} C_{8}(p)^{6}(q)^{6-6}$
$={ }^{6} C_{4}\left(\frac{1}{4}\right)^{5}\left(\frac{3}{4}\right)^{1}+{ }^{6} C_{0}\left(\frac{1}{4}\right)^{6}\left(\frac{3}{4}\right)^{0}$
$=\frac{1}{(4)^{6}}[6 \times 3+1]=\frac{19}{4096}$