A paint-store chain produces and sells latex and semigloss...

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?

Key Concepts

Conditional Probability

Conditional probability is the chance of an event occurring given that another event has already occurred. It is used to update the likelihood of outcomes based on known information and is fundamental when analyzing scenarios where the occurrence of one event affects the probability of another.

Law of Total Probability

The law of total probability is a principle that allows one to compute the probability of an event by breaking it down into several mutually exclusive cases. It combines the probabilities of the event across these distinct cases, often weighted by the likelihood of each case, to arrive at the overall probability.

Bayes' Theorem

Bayes' Theorem is a method for updating probabilities based on new evidence. It uses the relationship between conditional probabilities to reverse conditions, allowing one to determine the probability of a hypothesis given observed data, by combining prior probability with the likelihood of the observed evidence.

Transcript

0:00 Hello everyone.

00:01 So we will be going to solve problem number 101 from chapter 2 probability.

00:07 In this chapter it's being given that a person purchases two types of paint, latex and the semi -cloth paint.

00:15 And with the particular paint, he also purchases a ruler.

00:19 And the purchasing a roller depends on which paint he's purchasing.

00:22 Like there's different probabilities.

00:25 So there are two types of latex and semi -cloths.

00:31 So the probability of purchasing a latex paint is being given as 0 .75.

00:37 Purchasing of semi -cloth paint is 1 minus .75.

00:44 It becomes 0 .25 because there are only two types of paints that are available and person purchases one out of these two.

00:51 So the probability of latex purchasing is 0 .65.

00:54 So automatically the probability of semi -gloss becomes 0 .25.

01:01 Now, given that, he purpluses.

01:03 Purchases latex will he purchase a roller or not the probability is given as 0 .6 basically if he is purchasing a latex paint what is the probability of purchasing a roller this is the conditional probability we are talking about so that is 0 .6 this is being given in the question similarly if a person is purchasing a same gloss the probability of purchasing a roller is being given as 0 .3 now we have to tell that the what will the probability a person purchases a a roller and a paint...

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