Question

The least number of times a fair coin must be tossed so that the probability of getting at least one head in at least $0.8$ is:

• A. $3$
• B. $6$
• C. $5$
• D. none of these

Answer 1

Answer: (A)

Suppose the coin is tossed $n$ times. Let $x$ be the number of heads obtained.
Then $x$ follows a binomial distribution with parameters $n$ and $p=\frac{1}{2}$ ,
we have $p(x\ge 1)\ge 0.8\Rightarrow 1-P(x=0)\ge 0.8$
$\Rightarrow$ $P(x=0)\le 1-0.8=0.2$
$\Rightarrow$ ${}^n{C}_0{\left(\frac{1}{2}\right)}^n\le 0.2=\frac{2}{10}=\frac{1}{5}$
$\Rightarrow$ ${\left(\frac{1}{2}\right)}^n\le \frac{1}{5}\Rightarrow 2^n\le 5$
$\therefore$ Least value of $n$ is $3$ .