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

.: Let $p$ be the probability of success that coin falls heads $= \cfrac{1}{2}$

$\Rightarrow$ $p = \cfrac{1}{2},q = \cfrac{1}{2}$

$X$ has binomial distribution with $n = 20,p = \cfrac{1}{2},q = \cfrac{1}{2}$ .

$\therefore$ $P(X = r){ = ^n}{C_r}{(q)^{n - r}}{p^r}$

$\therefore$ Required probability $= P$ (he answers at least 12 answers correctly)

. $= P(X = 12) + P(X = 13) + P(X = 20)$

${ = ^{20}}{C_{12}}{\left( {\cfrac{1}{2}} \right)^8}{\left( {\cfrac{1}{2}} \right)^{12}} + \ldots \ldots \ldots \ldots { + ^{20}}{C_{20}}{\left( {\cfrac{1}{2}} \right)^{20}}$

$= {\left( {\cfrac{1}{2}} \right)^{20}}{[^{20}}{C_{12}}{ + ^{20}}{C_{13}} + \ldots \ldots \ldots { + ^{20}}{C_{20}}]$