Conditional probability given 2 scenarios

[Guest]

There are 4 books being sold in the bookshop : A, B, C, D.

We know that 20% of the male customers buy book A at least once a week, 55% buy book B at least once a week, 25% buy book C at least once a week and 15% buy book D at least once in a month.

We also know that 32% of the female customers by book A at least once a week, 80% buy book B at least once a week, 40% buy book C at least once a week and 65% buy book D at least once a week.

The ratio of male customers to female is 3 to 1.

The goal is to calculate a probability of meeting male and a female in the shop, given that each customer decided to purchase books A, B, C and the average frequency of shopping is once a week.



I believe the solution is to calculate joint probability of male and female probabilities of buying ABC set. Maybe I'm wrong so I could use some help. Also I'm not sure if shopping frequency matters.

[Guest]

Imagine 1000000 customers. "The ratio of male customers to female is 3 to 1" so there are 750000 male customers and 250000 female customers.

"We know that 20% of the male customers buy book A at least once a week, 55% buy book B at least once a week, 25% buy book C at least once a week and 15% buy book D at least once in a month."
Of the 750000 male customers, 20% of them, (0.2)(750000)= 150000, buy book A; 55% of them, (0.55)(750000)= 412500, buy book B; 25% of them, (0.25)(750000)= 187500, buy book C; and 15% of them, (0.15)(750000)= 112500, buy book C. (Was the "once a month" an error? Was it not "once a week" like all the others?)

"We also know that 32% of the female customers by book A at least once a week, 80% buy book B at least once a week, 40% buy book C at least once a week and 65% buy book D at least once a week."
So of the 250000 female custormers, (0.32)(250000)= 80000, buy book A; (0.80)(250000)= 200000, buy book B; (0.40)(250000)= 100000, buy book C; and (0.65)(250000)= 162500, buy book D.

"The goal is to calculate a probability of meeting male and a female in the shop, given that each customer decided to purchase books A, B, C and the average frequency of shopping is once a week."
I interpret this as meaning, "if a customer buys book A (or B, or C) what is the probability that customer is a man,"

There were 150000 males and 80000 females who bought book A. Of that total of 150000+ 80000= 230000 customers who bought book A, 150000 were males so, given that a custormer bought book A, the probability that customer is a male is [tex]\frac{150000}{230000}= 0.652[/tex] (rounded to 3 decimal places) and the probability that customer is a female is [tex]\frac{80000}{230000}= 0.348[/tex] (rounded to 3 decimal places). Of course, those add to 1.

The others are done the same way.