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# In an examination, 20 questions of true-false type 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.

Toolbox:
• For any Binomial distribution $B (n, p),$ 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 $(q = 1 – p)$
• Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions: (i) There should be a finite number of trials. (ii) The trials should be independent. (iii) Each trial has exactly two outcomes : success or failure. (iv) The probability of success remains the same in each trial.
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)$
edited Jun 21, 2013