Question

Rural and urban students are equally likely to get admission in a college. If $100$ students get admission, then the probability that more rural students get admission than urban students is

• A. ${\left(\frac{1}{2}\right)}^{100}{}^{100}{P}_{50}$
• B. ${\left(\frac{1}{2}\right)}^{100}{}^{100}{C}_{50}$
• C. $1-{\left(\frac{1}{2}\right)}^{100}{}^{100}{C}_{50}$
• D. $\frac{1}{2}(1-{\left(\frac{1}{2}\right)}^{100}{}^{100}{C}_{50})$

Answer 1

The correct option is D $\frac{1}{2}(1-{\left(\frac{1}{2}\right)}^{100}{}^{100}{C}_{50})$

Let $X=$ number of rural students
$p=q=\frac{1}{2}$
$\therefore P(X>50)$$={}^{100}{C}_{51}{\left(\frac{1}{2}\right)}^{51}{\left(\frac{1}{2}\right)}^{49}+.....+{}^{100}{C}_{100}{\left(\frac{1}{2}\right)}^{100}$
$=\frac{1}{2^{100}}({}^{100}{C}_{51}+{}^{100}{C}_{52}+...+{}^{100}{C}_{100})$
$=\frac{1}{2\times 2^{100}}(2{}^{100}{C}_{51}+2{}^{100}{C}_{52}+...+2{}^{100}{C}_{100})$
$=\frac{1}{2\times 2^{100}}({}^{100}{C}_0+{}^{100}{C}_1+...+{}^{100}{C}_{100}-{}^{100}{C}_{50})$
$=\frac{1}{2\times 2^{100}}(2^{100}-{}^{100}{C}_{50})$
$=\frac{1}{2}(1-{\left(\frac{1}{2}\right)}^{100}{}^{100}{C}_{50})$