Question

The least number of times a fair coin must be tossed so that the probability of getting at least one head is at least 0.8, is
(a) 7
(b) 6
(c) 5
(d) 3

Answer 1

(d) 3

Let X denote the number of coins.
Then, X follows a binomial distribution with

$$\begin{gathered} p=\frac{1}{2},q=\frac{1}{2} \\ \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}P(X\ge 1)\ge 0.8 \\ \Rightarrow 1-P(X=0)\ge 0.8 \\ \Rightarrow P(X=0)\le 1-0.8 \\ \Rightarrow P(X=0)=0.2 \\ \Rightarrow \frac{1}{2^n}\le 0.2 \\ \Rightarrow 2^n\ge \frac{1}{0.2} \\ \Rightarrow 2^n\ge 5 \\ \mathrm{This}\mathrm{is}\mathrm{possible}\mathrm{when}n\ge 3 \\ \mathrm{So},\mathrm{the}\mathrm{least}\mathrm{value}\mathrm{of}n\mathrm{is}3. \end{gathered}$$