Respuesta :
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
[tex]P(X|Z) = \dfrac{P(Z|X)(P(X) }{P(Z|X)*P(X)+P(Z|Y)*P(Y)}[/tex]
[tex]P(X|Z) = \dfrac{0.6 \times 0.75 }{(0.6 \times 0.75) +(0.3 \times 0.25)}[/tex]
[tex]P(X|Z) = \dfrac{0.45 }{(0.45) +(0.075)}[/tex]
[tex]P(X|Z) = \dfrac{0.45 }{0.525}[/tex]
[tex]P(X|Z) = 0.857[/tex]
Thus, the probability that the paint latex = 0.857
The concept of probability is used in order to determine the probability that the paint is latex and the probability is 0.857.
Given :
- The probability that a customer will purchase latex paint is 0.75.
- The customer who purchase latex paint, 60% also purchase rollers.
- The customer who purchase semigloss paint, 30% also purchase rollers.
First, determine the probability that a customer will purchase semigloss paint.
[tex]P(A) = 1- P(B)\\P(A) = 1 - 0.75\\P(A) = 0.25[/tex]
In the above expression P(B) is the probability that a customer will purchase latex paint.
Now, determine the probability that the paint is latex.
[tex]P(B|C) = \dfrac{P(C|B)\times P(B)}{P(C|B)\times P(B)+P(C|A)\times P(A)}[/tex]
Now, substitute the values of all the known terms in the above expression.
[tex]P(B|C) = \dfrac{0.6\times 0.75}{(0.6\times 0.75)+(0.3\times 0.25)}[/tex]
Simplify the below expression.
[tex]P(B|C) = \dfrac{0.45}{(0.45)+(0.075)}[/tex]
[tex]P(B|C) = \dfrac{0.45}{0.525}[/tex]
[tex]P(B|C) = 0.857[/tex]
So, the probability that the paint is latex is 0.857.
For more information, refer to the link given below:
https://brainly.com/question/795909
