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I'm reading a book Statistics: An Introduction from Roger E. Kirk. Now I'm studding the chapter about conditional probability and thought that I understood the examples well, and then this one came.

I'll quote the book:

Common Errors in Applying the Rules of Probability

...

$3$. Using the multiplication rule for statistically independent events $p(A \cap B)=p(A)p(B)$, when the events are not statistically independent. Suppose the probability of seeing the product, event A, is 0.40 and the probability of buying the product, event B, is 0.30. If the dependency between A and B is ignored, the incorrect probability of both seeing an advertisement and buying the product is $p(A\cap B)=0.40\times 0.30=0.12$. The correct probability takes in to account the conditional probability of buying the product given that the ad has been seen, $p(B|A)=0.50$, so that $p(A \cap B)=p(A)p(B|A)=0.40 \times 0.50=0.20$.

First I'm not sure if I understand the problem correctly. Does this mean that the probability of buying a product is 0 if he doesn't see the ad, and 0.3 if event A happened, that is if the ad was seen?

Or does this mean that the probability of buying the product is 0.3 by chance even if he didn't see it?

Although I don't understand what the problem exactly states, if I used the formulas given for finding $$p(B|A)=p(A\cap B)/p(A)$$ I would get $0.4\times 0.3/0.4=0.3$, how did he get $0.5$?

There are lot of questions already asked about conditional probability, but it didn't help to understand this problem and the result he got. Is there any ambiguity in this problem as is originally stated in the book?

EDIT: Since I can't comment, I'll put it here since I have no other way of interacting:
@Didier Piau yes, you're right, now that I see it I don't know why I wrote that.
@AMPerrine that is the source of my confusion I thought that he calculated p(B|A) somehow. Thank you for your answers.

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How do you know that p(A and B) is 0.4*0.3? This is not written in your book and in fact the whole purpose of the paragraph you quote is to emphasize that p(A and B) is not, in general, p(A)*p(B). –  Did Oct 28 '11 at 0:11
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He did not solve anything to get $p(B|A)=0.5$. That was a new piece of information introduced to the problem. I understand your confusion, as it came into play quite late. –  AMPerrine Oct 28 '11 at 0:19
    
It would seem that event A should be seeing an ad, not seeing the product. This is supported by the later calculation. –  Ross Millikan Oct 28 '11 at 4:49
    
@Quaid: You were unable to comment because you created a different account. I have now merged the accounts, so you will now able to comment here. However, until you reach 50 reputation points, you can only comment on your own questions and answers. –  Zev Chonoles Oct 28 '11 at 6:34
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2 Answers

The key point is to understand that the book is presenting a common error, specifically, to assume that the (two) variables are independent when, in fact, they are not. IF the variables are independent, then -- and only then -- one can simply multiply the probabilities using the multiplication rule (e.g., the probability of getting two heads with two tosses of a fair coin is $0.50 \times 0.50 = 0.25$). BUT, the point of the book paragraph is to warn readers NOT to use the multiplication rule if the variables are NOT independent; that is the common error. The author (of the book paragraph) then points out that buying the product is conditionally dependent on seeing the advertisement. That is, one must consider the probability of buying $B$ given that one has seen the advertisement $A$, which is written $p(B|A)$, and stated in plain English as "the probability of B given A." Next, the book author finally provides you with that information -- which is essential to solving the problem: i.e., that half the people who see the ad buy the product, so $p(B|A) = 0.50$. Then, using a variation of the axiom/definition of conditional probability given by the person who asked the question (the "variation" being that he rewrote it using simple algebra so there is no denominator), he calculates the correct answer which is $0.40 \times 0.50 = 0.20$. (The book explanation is not particularly clear if you didn't know the answer in the first place, so it's not your fault if you didn't follow it!)

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The mistake comes from the fact that there is relationship between buying a product and seeing the ad. Naturally, people are most likely to buy a product if he/she sees its ad. If you ignore the dependency and think they are independent, the result wouldn't reflect the situation in the real world. So in this case, the author assumes p(B|A)=0.5 and tries to calculate the value of p(A and B).

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