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 ______.

Answer

3

Explanation

`P(E_(1))=1-[P ("all heads)=P(all tails")]`
`=1-[(1)/(2^(n))+(1)/(2^(n))]=1-(1)/(2^(n-1))`
`P(E_(2))=["P(no head)+P (exactly one head)"]`
`=(1)/(2^(n))+""^(n)C_(1)xx(1)/(2^(n-1))=(n)/(2^(n))`
If `E_(1) and E_(2)` are independent, then
`(n)/(2^(n))(1-(1)/(2^(n-1)))((n+1)/(2^(n)))`
`orn =(1-(1)/(2^(n-1)))(n+1)`
`or n=n+1-(n+1)/(2^(n-1))`
`or n+1=2^(n-1)`
`or n=3`