The experiment is to toss a coin for each of the $n = 20 $ questions, and he coin falls heads, answer TRUE; if it falls tails, answer FALSE.

Let X be the number of correct answers. It is a case of Bernoulli trials as it satisfies the conditions (i) finite number of trials, (ii) independent trials, (iii) there is a definite outcome and (iv) the probability of success does not change for each trial.

P (a correct answer) $ = p = P (\text{Heads}) =\large \frac{1}{2}$; P (a wrong answer) $ = q = 1 - p = 1 - \large\frac{1}{2} = \frac{1}{2}$

Since X has a bionomial distribution, the probability of x success in n-Bernoulli trials, $P (X = x) = \large^{n}C_x. p^x.q^{n–x}$ where $x = 0, 1, 2,...,n$ and $(q = 1 – p)$

P (at least 12 correct answers) $= P (X=12) + P(X=13) + P (X=14)... P (X=20)$

$P (X = x) = \large^{20}C_x. \large\frac{1}{2}^x.\frac{1}{2}^{20–x} = \large^{20}C_x. \large\frac{1}{2}^{x+20-x} = \large^{20}C_x. \large\frac{1}{2}^{20}$

Therefore $ P (X \geq 12) =\large\frac{1}{2}^{20}$$ \times \sum_{12}^{20} \large(\large^{20}C_x)$