I have recently begun learning Probability Theory, and while I was working out the problems, I encountered this question
$60\%$ of the students applying for admissions are female. $30$ applications were received on a particular day. What is the probability that exactly $15$ of the applications will be from females? What is the probability that less than $10$ applications are female?
Is it Poisson Distribution of $Mean = 18$? How do I calculate the probability using Poisson Distribution?
The poisson distribution is usually used when trying to find the probability that a certain number of events happens in a fixed period of time.
This is a binomial distribution. Since the number of students applying is presumably very large, we can safely assume $p$ stays fixed, and won't increase or decrease based on previous observations. If $X\sim Binom(n,p)$ then
$$P(X=k)={n \choose k}p^k(1-p)^{n-k}$$
where in our case $n=30$ and $p=0.6$.
For example if you wanted to find the probability that exactly $15$ were female, you would take
$$P(X=15)={30 \choose 15}0.6^{15} 0.4^{15}\approx 0.0783$$
One could take two approaches to find the probability that less than $10$ applications are female.
Exact Probability Using Binomial Distribution:
$$\sum_{k=0}^9 {30 \choose k}0.6^{k} 0.4^{30-k}=0.000856392$$
However, without software, this could take a while so the alternative would be
Normal Approximation:
$$\begin{align*} P(X\lt 10) &=\Phi\left(\frac{X-\mu}{\sqrt{npq}}\right)\\\\ &=\Phi\left(\frac{9.5-18}{\sqrt{30\cdot0.6\cdot0.4}}\right)\\\\ &\approx0.0007680836 \end{align*}$$
Where I used continuity correction by finding $P(X\lt 9.5)$ as opposed to $P(X\lt 10)$