Question

How many times must a fair coin be tossed so that the probability of getting at least one head is more than $80\%$?

Answer 1

Let the required number of trials be $n$

Probablity of getting a head, $p=\frac{1}{2}$
Probability of getting a tail, $q=\frac{1}{2}$

According to the question:

P (at least one head) $>80\%$

$\Rightarrow P(x\ge 1)>\frac{8}{10}$

$\Rightarrow 1-P(x<1)>\frac{8}{10}$

$\Rightarrow P(x=0)<1-\frac{8}{10}$

$\Rightarrow {}^n{C}_0{q}^n{p}^0<\frac{2}{10}$

$\Rightarrow 1\times {\left(\frac{1}{2}\right)}^n<\frac{2}{10}$

$\Rightarrow {\left(\frac{1}{2}\right)}^n<\frac{1}{5}$ or $0.2$

$\Rightarrow {\left(\frac{1}{2}\right)}^n=\frac{1}{8},\frac{1}{16},\frac{1}{32},...$

$\therefore n=3,4,5,...$ will satisfy

Hence, $n\ge 3$