Marginal Distributions and PMF, I can't make sense of it and got it mixed up with conditional probability this example is not clear at all

Question

A company sells products in two categories, A and B. Some of its customers purchase products from both categories. These are the customers we are interested in. Category A has three products priced at \$100, \$300, and \$400. Category B has two products priced at \$100 and \$250. For a given customer, let the random variables X and Y be the prices of the products purchased from categories A and B, respectively. The following table gives the joint distribution of the customers who purchased a product from each of the two categories.

The probability that a randomly chosen customer purchases the \$100 product from category A and the \$250 product from category B is 2/12 ?? "how to calculate that". What is the probability that a randomly chosen customer purchases the \$100 product from category A? In this question, we have no mention of category B. In other words, we want the event where X = 100 and Y is "free".

I can't make sense of it and got it mixed up with conditional probability, I can't understand the table and the value of 12 as the total. when we state X=100, then the total should be 6, why 12 why are considering events when someone purchases both items from the same category? like how come the value of X=300 and Y=100 is 5/12???

Answer 1

The probability that a randomly chosen customer purchases the \$100 product from category A and the \$250 product from category B is 2/12 ?? "how to calculate that".

You read it off the table. The cell in the column of $X=100$ and row of $Y=250$ contains $2/12$. That is the probability for that event.

What is the probability that a randomly chosen customer purchases the $100 product from category A? In this question, we have no mention of category B. In other words, we want the event where X = 100 and Y is "free".

Sum the contents of all cells in the column of $X=100$. It is the Law of Total Probability.

I can't make sense of it and got it mixed up with conditional probability, I can't understand the table and the value of 12 as the total.

The sum of all cells in the table must equal 1, and it does.

when we state X=100, then the total should be 6, why 12

There are six cells. That does not mean their contents need to be integer multiples of $1/6$. There is no reason for that.

There are six cells because that is the number of choices to be made when selecting one item in each category, when there are 3 choices in category A and 2 choices in category B.

But perhaps we polled 12 customers who purchase one item in each category.

why are considering events when someone purchases both items from the same category?

We are considering the distribution of purchases for people who purchase one item in each category. This is just what we are measuring. This is not generated by a mathematical model, but on statistics gathered on the choices of such customers.

like how come the value of X=300 and Y=100 is 5/12???

The polls show that an average of 5 in 12 such customers chose to purchase a \$300 priced item from category A and a \$100 priced item from category B.

It is just a popular choice among customers who purchase one item in each category. Who knows why.