Probability of hitting the target for the nth time on the mth throw A boy is throwing stones at a target. Probability of him hitting the target is $\frac{1}{2}$ . Then the probability of him hitting the target for the  5th time in 10 throws is ?
The answer given is 1/2 .

Now ,the reason given is that no matter how many throws he takes or no matter how many times he hits (be it 10 times in 10 throws or 6 times in 7 throws or all initial 5 throws hitting the target out of 10 throws ) since the probability of him hitting the target is always 1/2 therefore the net probability will always be 1/2.
But while putting it into the binomial distribution formula the answer coming is different.

Probability of hitting target is $~\frac{1}{2}~$ and therefore probability of hitting target is $~1- \frac{1}{2}=\frac{1}{2}~$.
So $~P~=~$$\{$ probability of hitting target $~4~$times in $~9~$ throw $\}~-~\{$ probability of hitting target in $~10^{th}~$ throw $~\}$
$~~~~~~~~~~=~\left\{^9C_4 ~\times~ \left(\frac{1}{2}\right)^4~\times~ \left(\frac{1}{2}\right)^5\right\}~\times~\left(\frac{1}{2}\right)$
$~~~~~~~~~~=~^9C_4 ~\times~ \left(\frac{1}{2}\right)^{10}$
$~~~~~~~~~~=\frac{63}{2^9}$
