Expert: Paul Klarreich - 11/27/2009

Question
Hello Paul,

I just came across your profile on allexperts.com. I have been struggling with a probability question for quite some time now. Here it goes
"A management consultancy firm offers a sales aptitude testing service. It is claimed that when this test is applied to a group of “top-class” salespeople, 95% pass and 5% fail.  When it is applied to a group of “average” sales people, 10% pass and 90% fail.
The company considering use of this service believes that only 1 in 10 of salespeople interviewed has “top-class” potential, and that its current selection procedure results in a mix of 60% “top-class” and 40% “average” salespeople.

(a)   if a salesperson passes the test, what is the probability that he or she is “top-class”? [i.e. what is the posterior probability P(“top-class” | pass test)]?

(b)   Is this a good test for discriminating the potential quality of salespeople?

Now, what I know is the formula for P(A|B) but what I am not sure is why they have given the details about mix of the candidates selected (60/40). I know that this would affect the result but not sure how to use this information. Any pointers to correct solution would be a great help.

Best Regards,
Mike

Answer
Questioner: Mike
Country: India
Category: Advanced Math
Private: No
Subject: Conditional Probability Question
Question: Hello Paul,

I just came across your profile on allexperts.com. I have been struggling with a probability question for quite some time now. Here it goes
"A management consultancy firm offers a sales aptitude testing service. It is claimed that when this test is applied to a group of “top-class” salespeople, 95% pass and 5% fail.  When it is applied to a group of “average” sales people, 10% pass and 90% fail.
The company considering use of this service believes that only 1 in 10 of salespeople interviewed has “top-class” potential, and that its current selection procedure results in a mix of 60% “top-class” and 40% “average” salespeople.

(a) if a salesperson passes the test, what is the probability that he or she is “top-class”? [i.e. what is the posterior probability P(“top-class” | pass test)]?

(b) Is this a good test for discriminating the potential quality of salespeople?

Now, what I know is the formula for P(A|B) but what I am not sure is why they have given the details about mix of the candidates selected (60/40). I know that this would affect the result but not sure how to use this information. Any pointers to correct solution would be a great help.

Best Regards,
Mike
..................................
Let's try to remove some of the excess verbosity.

Let P = 'the candidate passes'
   T = 'the candidate is top-class'

Then:

It is claimed that when this test is applied to a group of “top-class” salespeople, 95% pass and 5% fail.

means:

p(P / T) = 0.95


When it is applied to a group of “average” sales people, 10% pass and 90% fail.
(Assumption:  "average" means "not topclass".)

means:

p(P / -T) = 0.10


The company considering use of this service believes that only 1 in 10 of salespeople interviewed has “top-class” potential,

means:

p(T) = 0.10, so  p(-T) = 0.9


its current selection procedure results in a mix of 60% “top-class” and 40% “average” salespeople.

Let C = 'salesman passes current selection procedure'

then p(T / C) = 0.60

.....................................
OK, let's apply some rules:

p(P / T) = p(P T)/p(T)

0.95 = p(PT)/0.10

p(P T) = 0.095    << first conclusion.

p(P / -T) = p(P -T)/p(-T)

0.10 = p(-T P) /0.9

p(P -T) = 0.09

Now

p(P) = p(P T) + p(P -T)

p(P) = 0.095 + 0.09 = 0.185



(a) if a salesperson passes the test, what is the probability that he or she is “top-class”? [i.e. what is the posterior probability P(“top-class” | pass test)]?

means

compute  p(T / P) = p(P T)/p(P)

p(T / P) = 0.095/0.185  = 95/185 = 19/37


(b) Is this a good test for discriminating the potential quality of salespeople?


I think this means:  If the 'test' is applied to some group and you only hire those who pass, then some 19/37 of your staff will be topclass.  This is slightly more than 50%, but right now you get about 60%.

You draw a conclusion, now.