Question

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

Answer 1

Let X denote the number of correct answers given by the student. 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 of the question are Bernoulli trails.

$\therefore$ p=P(a success) = P (coin show up a head) = $\frac{1}{2}$

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

$\therefore$ X has binomial distribution with n = 20, $p=\frac{1}{2}andq=\frac{1}{2}$

$P(X-r)={}^{20}{C}_r.{\left(\frac{1}{2}\right)}^{20-r}=P(X\ge 12)=P\left(12\right)+P\left(13\right)+P\left(14\right)+P\left(15\right)+P\left(16\right)+P\left(17\right)+P\left(18\right)+P\left(19\right)+P\left(20\right)={}^{20}{C}_{12}{p}^{12}{q}^8+{}^{20}{C}_{13}{p}^{13}{q}^7+{}^{20}{C}_{14}{p}^{14}{q}^6+{}^{20}{C}_{15}{p}^{15}{q}^5+{}^{20}{C}_{16}{p}^{16}{q}^4+{}^{20}{C}_{17}{p}^{17}{q}^3+{}^{20}{C}_{18}{p}^{18}{q}^2+{}^{20}{C}_{19}{p}^{19}{q}^1+{}^{20}{C}_{20}{p}^{20}{q}^{20}={(}^{20}{C}_{12}{+}^{20}{C}_{13}{+}^{20}{C}_{14}{+}^{20}{C}_{15}{+}^{20}{C}_{16}{+}^{20}{C}_{17}{+}^{20}{C}_{18}{+}^{20}{C}_{19}{+}^{20}{C}_{20})\frac{1}{2^{20}}={\left(\frac{1}{2}\right)}^{20}{[}^{20}{C}_{12}{+}^{20}{C}_{13}+...{+}^{20}{C}_{20}]$