Question

An unbiased coin is tossed $n$ times. Let $E_1$ be the event that both heads and tails are present in $n$ tosses. Let $E_2$ be the event that the coin shows up heads at most once in $n$ tosses. The value of $n$ for which $E_1$ and $E_2$ are independent is

Answer 1

$n\left(S\right)=2^n$
$n\left(E_1\right)=2^n-2$
$n\left(E_2\right)={}^n{C}_1+1=n+1n(E_1\cap E_2)=n$
$\because P(E_1\cap E_2)=P\left(E_1\right)\times P\left(E_2\right)\Rightarrow \frac{n}{2^n}=\frac{2^n-2}{2^n}\times \frac{n+1}{2^n}\Rightarrow n=3$