1. ## Impurities probability problem

Hi guys! I'm new in theory of probability and I'm having a hard time solving a probability problem. I'll be really grateful if you explain to me the solution.
So here's the problem:

Test for impurities commonly found in drinking water from private wells showed that 30% of all wells in a particular country have impurity A. If a random sample of five wells is selected from the large number of wells in the country, what is the probability that:

a) Exactly three will have impurity A?
b) At least three?
c) Fewer than three?

2. For a)

There are $\binom{5}{3}$ ways to select exactly 3 wells from the sample of 5.

Therefore the probability is $\binom{5}{3}0.3^30.7^2$

For b)

At least 3 means 3, 4 or 5...

The probability of 4 having impurity A is $\binom{5}{4}0.3^40.7$

The probability of 5 having A is $0.3^5$

The sum of these 2 probabities and the answer to a) is the answer to b)

For c), just find the probability of 0, 1 or 2 having A, or subtract the answer for b) from 1