probability problem in statistic

Question

A study found that 25% of the bottled water from different brands is not safe to drink. Suppose a researcher randomly selects 20 bottled water from different brands. let x is the number of bottled water from different brands which is not safe to drink. 

a. Explain why x is a binomial random variable. b. What is the probability that at least one bottled water is not safe to drink? c. What is the probability that exactly 18 bottled water is safe to drink? d. Find P(x < 2). Interpret the result. e. What is the average no. of bottled water which is not safe to drink?  

Answer 1

(a) because there are 2 possibilities for the outcome - the water is fit to drink, or not

(b) Probability all water is safe to drink = 0.25²⁰

Probability (some of water not safe) = 1 - 0.25²⁰

(c) We can choose the 18 drinks ok in C (20,18) ways.

For a given way, probability is 0.75¹⁸ 0.25²

So total probability is C (20,18) 0.75¹⁸ 0.25²

(d) p (x<2) = p (x=0) + p (x=1) = 0.75²⁰ + C (20,1) 0.75¹⁹ 0.25¹

(e) Average number will be 0.25 x 20 = 5