Please help me to find the probability of this question

Question

Suppose that about 85% of graduating students attend their graduation. A group of 21 graduating students is randomly chosen.

Find the probability that 15 or 16 attend.

Answer 1

This is a binomial probability question. You can know it is a binomial probability because

1. There is a fixed probability p of 85%. This probability does not change based on n and represents a binary option: attend graduation or don't.
2. There is a fixed n number of trials of 21.
3. There is a fixed x of either 15 or 16.

For a binomial the probability P(x)=(n!/((x!)*(n-x)!))(p^x)((1-p)^(n-x)). Since we are wondering about 15 or 16 you need to try each of these values separately for x.

Thus for 15, P(X)=(21!/(15!*(21-15)!))(0.85¹⁵)((1-0.85)^(21-15)=0.05399

For 16 using the same formula, the probability is 0.11474.

The events that 15 and 16 people attend are considered mutually exclusive because there is not a scenario where both 15 people attend AND 16 people attend (it must be one or the other). Therefore, these two probabilities can be added together for the final probability.

The probability that 15 or 16 attend is 0.05399+0.11474=0.16873 or 16.873%.