Bayes' Rule

Question

Plantation Foods has 710 employees. A total of 130 employees have a college degree, and 580 do not have college degrees. Of those with college degrees, 60% are men and 40% are women. Of those who do not have college degrees, 25% are men and 75% are women. The Human Resources Office selects an employee at random to interview about a proposed health insurance change. The person selected is a woman. Find the probability that she does not have a college degree. (Round your answer to three decimal places.)

Answer 1

Bayes' Rule of Conditional Probability:

 

                   P(B|A)*P(A)

P(A|B) =  -------------------

                       P(B)

 

Where:

• Proposition A = The person has no college degree
• Proposition B = the person randomly selected is a woman.  

 

Hence we a re looking for the probability that the person selected has no college degree given that the person randomly selected is a woman.

 

First let's crunch some numbers.  There are 710 employees:

• 130 employees have a college degree, 580 do not
• 130*0.4 = 52 women have a college degree
• 580*0.75 = 435 women do not have a college degree
• There are a total of 52+435 = 487 woman at the company

 

Now let's figure out P(B|A).  This is the probability that the person chosen is a woman given that the chosen person has no college degree.  There are 580 employees without a college degree, 435 of whom are women.  Hence P(B|A) = 435/580

 

P(A) is the probability that the person chosen does not have a college degree.  There are 710 total employees, 580 of whom do not have a college degree.  Hence P(A) = 580/710

     

P(B) is the probability that the randomly chosen person is a woman.  There are 710 employees, 487 of whom are women.  Hence P(B) = 487/710

 

Now we're ready to use Bayes' Rule:

 

                   (435/580)*(580/710)

P(A|B) =  --------------------------------- = 435/487 = 89.3%

                         (487/710)

 

Of course you could have done this more directly by taking the ratio of women without a college degree (435) to (over) the total number of women (487).