Answer to Question #143758 in Statistics and Probability for qwerty

Question

3. A paint-store chain produces and sells latex and semigloss paint. Based on long-range sales among these two paints, the probability that a customer will purchase latex paint is 0.75. Of those that purchase latex paint, 60% also purchase rollers. But only 30% of semigloss paint buyers purchase rollers. A randomly selected buyer purchases a roller and a can of paint. What is the probability that the paint is latex?

(a) What is the probability that a customer will purchase a roller? (Give answer in decimal form in thousandths place. Ex. 0.031)
(b) What is the probability that the paint is semigloss? (Give answer in simplified fraction form. Ex. 31/51)
(c) What is the probability that the paint is latex? (Give answer in simplified fraction form. Ex. 31/51)

Answer 1

We assume that every buyer purchases a paint. I.e., the probability that a buyer purchases a latex paint is 0.75 and the probability that a buyer purchases a semigloss paint is 0.25. We denote by $A$ the event: "a buyer purchases a roller" and by $B$ the event: "a buyer purchases a latex". The aim is to find $P(B|A)$. The conditional probability is given by (see https://en.wikipedia.org/wiki/Conditional_probability) $P(B|A)=\frac{P(B\cap A)}{P(A)}$ .

$P(B\cap A)=0.75\cdot0.6=0.45$; $P(A)=0.75\cdot0.6+0.25\cdot0.3=0.525$ . In order to calculate the latter, we used a law of total probability (https://en.wikipedia.org/wiki/Law_of_total_probability). Thus, $P(B|A)=\frac{0.45}{0.525}\approx0.8571$

(a) The probability that a buyer will purchase a roller is: $0.75\cdot0.6+0.25\cdot0.3=0.525$ It is obtained using a law of total probability (see https://en.wikipedia.org/wiki/Law_of_total_probability).

(b) Due to the assumption that we made at the beginning, the answer is $0.25=\frac{1}{4}$

(с) Due to the assumption that we made at the beginning, the answer is $0.75=\frac{3}{4}$