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?

  • $\begingroup$ Where does the Poisson distribution come out? Isn't this just counting numbers? $\endgroup$ – Saad Mar 13 '18 at 5:50
  • $\begingroup$ I don't know how to do this answer. Given that 60% is the average is established, shouldn't I calculate what is the probability that it is 50%? $\endgroup$ – Anirudh Murali Mar 13 '18 at 5:53

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 $P(X\lt 10)$

  • $\begingroup$ How do I differentiate between Poisson and Binomial, seems very confusing? Is there a ELI5 for this? $\endgroup$ – Anirudh Murali Mar 13 '18 at 6:12
  • $\begingroup$ The poisson distribution is usually used when dealing with time. For instance, if we know that something happens $3$ times per hour, on average, then we can use this to find the probability that, say, $5$ events happen in $2$ hours. $\endgroup$ – Remy Mar 13 '18 at 6:15

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