Question

In a 20-question true-false examination, suppose a student tosses a fair coin to determine his answer to each question. For every head, he answers 'true' and for every tail, he answers 'false'. Find the probability that he answers at least 12 questions correctly.

Answer 1

Let X denote the number of correct answers.
Then, X follows a binomial distribution with $n\mathit{}=20$

$$\begin{gathered} \text{Let}\mathit{}p\mathrm{be}\mathrm{the}\mathrm{probability}\mathrm{of}\text{a correct}\mathrm{answer}. \\ \Rightarrow p=\mathrm{getting}\mathrm{a}\mathrm{head}\mathrm{and}\text{a}\mathrm{right}\mathrm{answer}\mathrm{to}\mathrm{be}\mathrm{true}\mathrm{or}\mathrm{getting}\mathrm{a}\mathrm{tail}\mathrm{and}\text{a}\mathrm{right}\mathrm{answer}\mathrm{to}\mathrm{b}e\mathrm{false} \\ \Rightarrow p=\frac{1}{2} \\ \therefore q=1-\frac{1}{2}=\frac{1}{2} \\ \mathrm{Hence},\mathrm{the}\mathrm{distribution}\mathrm{is}\mathrm{given}\mathrm{by} \\ P(X=r)={}^{20}\mathit{\text{C}}_r{\left(\frac{\mathit{1}}{\mathit{2}}\right)}^r{\left(\frac{\mathit{1}}{\mathit{2}}\right)}^{20-r},r=0,1,2,3......20 \\ =\frac{{}^{20}\mathit{\text{C}}_r}{2^{20}} \\ \mathrm{Probability}\mathrm{that}\text{the student}\mathrm{answers}\mathrm{at}\mathrm{least}12\mathrm{questions}\mathrm{correctly}=P(X\ge 12) \\ =P(X=12)+P(X=13)+...+P(X=20) \\ =\frac{{}^{20}\mathit{\text{C}}_{\mathit{12}}+{}^{20}\mathit{\text{C}}_{13}+...+{}^{20}\mathit{\text{C}}_{\text{20}}}{2^{20}} \end{gathered}$$