Question

In a exam there are $30$ true/false questions. If a student guesses all the $30$ questions, then the probability that he/she gets atleast $15$ correct, is

• A. $\frac{1}{2}$
• B. $1-\frac{{}^{30}{C}_{15}}{2^{30}}$
• C. $\frac{1}{2}+\frac{{}^{30}{C}_{15}}{2^{31}}$
• D. $\frac{1}{2}+\frac{{}^{30}{C}_{15}}{2^{30}}$

Answer 1

The correct option is C $\frac{1}{2}+\frac{{}^{30}{C}_{15}}{2^{31}}$

Probability that a question is correct
$P\left(C\right)=\frac{1}{2}$
Probability that a question is incorrect
$P\left(IC\right)=\frac{1}{2}$
Now, probability that he/she get atleast $15$ questions correct
$P\left(E\right)=\frac{[{}^{30}{C}_{15}+{}^{30}{C}_{16}+{}^{30}{C}_{17}+\dots +{}^{30}{C}_{30}]}{2^{30}}\cdots \left(i\right)\Rightarrow P\left(E\right)=\frac{[{}^{30}{C}_{15}+{}^{30}{C}_{14}+{}^{30}{C}_{13}+\dots +{}^{30}{C}_0]}{2^{30}}\cdots \left(ii\right)$

Adding equations $\left(i\right)$ and $\left(ii\right)$, we get
$\Rightarrow 2P\left(E\right)=\frac{[{}^{30}{C}_0+{}^{30}{C}_1+\dots +{}^{30}{C}_{30}+{}^{30}{C}_{15}]}{2^{30}}\Rightarrow 2P\left(E\right)=\frac{[2^{30}+{}^{30}{C}_{15}]}{2^{30}}\therefore P\left(E\right)=\frac{1}{2}+\frac{{}^{30}{C}_{15}}{2^{31}}$