Question

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

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is C 3

Let's denote head by 'H' and tails by 'T'
$P\left(E_1\right)=1-\{P\left(allTs\right)+P\left(allHs\right)\}=1-\{\frac{1}{2^n}+\frac{1}{2^n}\}=1-\frac{1}{2^{n-1}}$
so $E_2=(n-1)Tand1HornT\Rightarrow P\left(E_2\right)=\left(\begin{array}{c}n\\ 1\end{array}\right){0.5}^1\times {0.5}^{n-1}+{0.5}^n\Rightarrow P\left(E_2\right)=\frac{n+1}{2^n}$
It can be seen very clearly that $E_1\cap E_2=(n-1)Tand1H$

$\Rightarrow P(E_1\cap E_2)=\left(\begin{array}{c}n\\ 1\end{array}\right){0.5}^1\times {0.5}^{n-1}=\frac{n}{2^n}$

now for independence $P(E_1\cap E_2)=P\left(E_1\right)P\left(E_2\right)\Rightarrow \frac{n}{2^n}=(1-\frac{1}{2^{n-1}})\left(\frac{n+1}{2^n}\right)\Rightarrow n\times 2^{n-1}=(n+1)(2^{n-1}-1)\Rightarrow 2^{n-1}-n-1=0\Rightarrow n=3$