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

Question:

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.

Solution:

Let X represent the number of correctly answered questions out of 20 questions.

The repeated tosses of a coin are Bernoulli trails. Since “head” on a coin represents the true answer and “tail” represents the false answer, the correctly answered questions are Bernoulli trials.

$\therefore p=\frac{1}{2}$

$\therefore q=1-p=1-\frac{1}{2}=\frac{1}{2}$

$X$ has a binomial distribution with $n=20$ and $p=\frac{1}{2}$

$\therefore \mathrm{P}(\mathrm{X}=x)={ }^{9} \mathrm{C}_{x} q^{n-x} p^{x}$, where $x=0,1,2, \ldots n$

X has a binomial distribution with $n=20$ and $p=\frac{1}{2}$

$\therefore \mathrm{P}(\mathrm{X}=x)={ }^{n} \mathrm{C}_{x} q^{n-x} p^{x}$, where $x=0,1,2, \ldots n$

$={ }^{20} \mathrm{C}_{x}\left(\frac{1}{2}\right)^{20-x} \cdot\left(\frac{1}{2}\right)^{x}$

$={ }^{20} \mathrm{C}_{x}\left(\frac{1}{2}\right)^{20}$'

$={ }^{20} \mathrm{C}_{x}\left(\frac{1}{2}\right)^{20}$

P (at least 12 questions answered correctly) = P(X ≥ 12)

$=\mathrm{P}(\mathrm{X}=12)+\mathrm{P}(\mathrm{X}=13)+\ldots+\mathrm{P}(\mathrm{X}=20)$

$={ }^{20} \mathrm{C}_{12}\left(\frac{1}{2}\right)^{20}+{ }^{20} \mathrm{C}_{13}\left(\frac{1}{2}\right)^{20}+\ldots+{ }^{20} \mathrm{C}_{20}\left(\frac{1}{2}\right)^{20}$

$=\left(\frac{1}{2}\right)^{20} \cdot\left[{ }^{20} \mathrm{C}_{12}+{ }^{20} \mathrm{C}_{13}+\ldots+{ }^{20} \mathrm{C}_{20}\right]$