Question

In an examination,20 questions of true-false are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads ,he answers 'true'. if it falls tails, he answers 'false'. Find the probability that he answers at least 12 questions correctly.

Answer 1

S={H,T} $\Rightarrow$ n(S)=2

Let A represents a head.

∴ A={H} $\Rightarrow$ n(A)=1

$p= \frac{n(A)}{n(S)}=\frac{1}{2}\, and \; q=1-p=1-\frac{1}{2}=\frac{1}{2}$

n=20,r=12,13,...20 and P(X=r)=$^nC_rp^rq^{n-r}$

P (at least 12 success)=P(X=12)+P(X=13)+....P(X=20)

=$^{20}C_{12}\bigg(\frac{1}{12}\bigg)^{12}\bigg(\frac{1}{2}\bigg)^8+^{20}C_{13}\bigg(\frac{1}{2}\bigg)^7+...^{20}C_{20}\bigg(\frac{1}{2}\bigg)^{20}\bigg(\frac{1}{2}\bigg)^0$

=$\bigg(\frac{1}{2}\bigg)^{20}\bigg[^{20}C_{13}+^{20}C_{14}+^{20}C_{15}+^{20}C_{16}+^{20}C_{17}+^{20}C_{18}+^{20}C_{19}+^{20}C_{20}\bigg]$