Suppose that 30% of all small businesses are undercapitalized. 40% of all undercapitalized small businesses fail and 20% of all small businesses that are not undercapitalized fail. A small business is chosen at random. The probability that the small business succeeds if it is undercapitalized is?
Greg
Hi Greg,
This is a conditional probability question. You are asked the probability that the business succeeds given that it is underfunded.
I would use the expression for conditionla probability to do this. If A and B are events then the probability of A given B is
Pr(A|B) = Pr(A and B)/Pr(B)
In your problem A is the event that the business succeeds
and B is the event that the business is underfunded. Thus Pr(B) = 0.30 and
Pr(A and B) = 0.40 
0.30 = 0.12.
Thus
Pr(A|B) = 0.12/0.3 = 0.4
I think a much more interesting question is "If you randomly select a small business, what is the probability that it is underfunded if it is successful?"
Penny
In May of 2007 we received this note from Sarah
Hi,
I ran across this page in the Quandaries and Queries section of your website.
The solution is incorrect. The problem statement says that "40% of all undercapitalized small businesses fail." The solution then uses A as "the event that the business succeeds," and uses .40 for that value. It should instead use 0.60 for the success value.
Sarah is correct, my previous answer is wrong. The question says "40% of all undercapitalized small businesses fail" so "60% of all undercapitalized small businesses fail"
Thanks Sasah,
Penny