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I am looking for some help on understanding how to approach this problem / some intuition. It is a question from an old exam. I know the answer to (a) is 63%, and the answer to (b) is $\frac {0.21}{0.28}=0.75$. Its for an introductory statistics for economics course, and the only problems I seem to be struggling with are those involving probability.

Suppose that 30% of independent bookstores are profitable. There is a 70% probability that, if it is profitable, an independent bookstore will be taken over by a larger chain. Among non-profitable, independent bookstores only 10% are taken over.

(a) What is the probability that an independent bookstore is not profitable and not taken over? (b) If you observe a takeover, what is the probability that the store in question was profitable?

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4 Answers

up vote 1 down vote accepted

There are four possible cases:

profitable ($30$%) and taken over ($70$%) = $21$%

profitable ($30$%) and not taken over ($30$%) = $9$%

not profitable ($70$%) and taken over ($10$%) = $7$%

not profitable ($70$%) and not taken over ($90$%) = $63$%

The number on the right of the = is those two probabilities multiplied together. Notice how all four results add up to $100$%. For problem (a), the answer is $63$% because it is simply the product of the two probabilities.

For problem (b), you have to ignore some data. Out of the $28$ percent chance that it is taken over (the sum of the two cases), $21$ percent is in the case of it being profitable. This is $75$% of the total, which is the answer.

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What you need to do first is identify what events are being discussed, and translate the data and the question from English to mathematics. In this one the events are

the bookstore is profitable (call this $A$)

the bookstore is taken over (call this $B$)

You are given that $P(A) = 0.30$, that $P(B|A) = 0.70$, and that $P(B|A^c) = 0.1$, and you are asked for $P(A B^c)$ and $P(A|B)$.

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For (a): $70\% $ of the independent book stores are non-profitable and $90\%$ of those won't be taken over by a larger chain. Therefore $90\%\times 70\%=63\%$ is the probability that an independent bookstore is not profitable and not taken over. For (b): $40\%$ of the independent book stores will be taken. Of those 3 out of 4 will be from a profitable store. Thus there is a $3/4=75\%$ if probability that it is a profitable one.

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What I do intuitively is partition the space of possibilities first naively and then refine this partition using additional information. Here is a graphical depiction of what goes on in my head:

$~~~$ partition

Circled in gray are the pieces of information we deduce from disjointness (if two events, like taken over and not taken over, are disjoint but exhaust all possibilities, their probability adds up to one-hundred percent). After we do this partitioning we can start applying "of" to the smaller pieces:

  • 70% of 30% = 21% of indie bookstores are profitable and taken over
  • 30% of 30% = 9% of indie bookstores are profitable and not taken over
  • 10% of 70% = 7% of indie bookstores are not profitable and taken over
  • 90% of 70% = 63% of indie bookstores are not profitable and not taken over

We can now create a table that exhibits greater detail of what is going on:

$~~$ table

From the above information we can deduce:

  • (21%)/(28%) = 75% of taken over bookstores are profitable
  • (7%)/(28%) = 25% of taken over bookstores are not profitable
  • (9%)/(72%) = 12.5% of not taken over bookstores are profitable
  • (63%)/(72%) = 87.5% of not taken over bookstores are not profitable
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