Question

A paint-store chain produces and sells latex and semigloss paint. Based on long-range sales, 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?

Answer 1

Answer:

The probability that the paint  latex = 0.857

Step-by-step explanation:

From the given information:

Let consider X to be the event that the customer has purchased the latex paint

Let consider Y to be the event that the customer has purchased semigloss paint.

Now, the probability of X i.e. P(X) = 0.75

SInce X and Y are mutually exhaustive)

P(Y) = 1 - P(X)

P(Y) = 1 - 0.75

P(Y) = 0.25

Similarly, Let Consider Z be the event that the customer has purchased a roller.

We are being told that P(Z|X) = 0.6

P(Z|Y) = 0.30

$P(X|Z) = \dfrac{P(Z|X)(P(X) }{P(Z|X)*P(X)+P(Z|Y)*P(Y)}$

$P(X|Z) = \dfrac{0.6 \times 0.75 }{(0.6 \times 0.75) +(0.3 \times 0.25)}$

$P(X|Z) = \dfrac{0.45 }{(0.45) +(0.075)}$

$P(X|Z) = \dfrac{0.45 }{0.525}$

$P(X|Z) = 0.857$

Thus, the probability that the paint  latex = 0.857