# Math Help - Binomial distribution

1. ## Binomial distribution

This problem have been taken from the old paper of MASTER of COMMERCE Examination of well-known University in India.
The probability of a man hitting a target is $\frac14.$How many times must he fire so that the probability of his hitting the target at least once is greater than $\frac23$ ?

3.818 times i-e approximately 4 times
A man must fire 4 times so that the probability of his hitting the target at least once is greater than $\frac23$
Answer is not available in the university paper set.So I don't know whether I solved the problem wrongly or correctly.

2. ## Re: Binomial distribution

Originally Posted by Vinod
This problem have been taken from the old paper of MASTER of COMMERCE Examination of well-known University in India.
The probability of a man hitting a target is $\frac14.$How many times must he fire so that the probability of his hitting the target at least once is greater than $\frac23$ ?

3.818 times i-e approximately 4 times
A man must fire 4 times so that the probability of his hitting the target at least once is greater than $\frac23$
By definition it must be $(\frac{3}{4})^n<\frac{1}{3} \implies (\frac{4}{3})^n>3 \implies n\ \ln \frac{4}{3}>\ln 3 \implies$
$\implies n> \frac{\ln 3}{\ln \frac{4}{3}} = 3.8188416793...$
$\chi$ $\sigma$