using bayes theorem

Question

Fora hypothetical magazine, the probability that the reader is male given the reader is at least 35 years old is 0.30. The probability that the reader is male, given that the reader is under 35 is 0.65. If 75% of the readers are under 35, what is the probability that a randomly chosen reader is under 35 if it is given the reader is a female?

Answer 1

65% of the readers under the age of 35 are male.

75% of the total readers are under the age of 35.

 

Now it's important to pay attention to what you are solving for.  We are not solving for the odds that a female is chosen out of the whole bunch.  We're saying that a female has already been chosen.  The only question is what percentage of the women are under 35.  

 

We can solve this by assuming we are working with a group of 100 people.  We will then figure out how many of the 100 are women.  Then we can figure out what percentage of the women are under 35.

 

So assume you have 100 readers.

75 of them are under 35.

65% of that 75 are male.

   .65*75 = 48.75.

   48.75/75 of those under 35 are male.

   26.25/75 of them are female.

 

Of the 25 who are over 35, 70% of them are female.

    .70*25 = 17.5

    17.5/25 are female

 

So you have 43.75 females total out of the 100.

    26.25/43.75 females are under 35, or 60% of all females.

 

If a female is randomly chosen, there is a 60% chance she is under 35.

 

Hope this helps Sarah!