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,
$P(X=5)+P(X=6)$
$={}^6{C}_5(p{)}^5(q{)}^{6-5}+{}^6{C}_8(p{)}^6(q{)}^{6-6}$
$={}^6{C}_4{\left(\frac{1}{4}\right)}^5{\left(\frac{3}{4}\right)}^1+{}^6{C}_0{\left(\frac{1}{4}\right)}^6{\left(\frac{3}{4}\right)}^0$
$=\frac{1}{(4{)}^6}[6\times 3+1]=\frac{19}{4096}$