Saturday, January 21, 2006

Fun Quiz!! What's wrong with this?

Conditional probabilities
To illustrate, suppose there are two bowls full of cookies. Bowl #1 has 10 chocolate chip cookies and 30 plain cookies, while bowl #2 has 20 of each. Our friend Fred picks a bowl at random, and then picks a cookie at random. We may assume there is no reason to believe Fred treats one bowl differently from another, likewise for the cookies. The cookie turns out to be a plain one. How probable is it that Fred picked it out of bowl #1?

Intuitively, it seems clear that the answer should be more than a half, since there are more plain cookies in bowl #1. The precise answer is given by Bayes' theorem. But first, we can clarify the situation by rephrasing the question to "what’s the probability that Fred picked bowl #1, given that he has a plain cookie?” Thus, to relate to our previous explanation, the event A is that Fred picked bowl #1, and the event B is that Fred picked a plain cookie. To compute Pr(A|B), we first need to know:

Pr(A), or the probability that Fred picked bowl #1 regardless of any other information. Since Fred is treating both bowls equally, it is 0.5.

Pr(B), or the probability of getting a plain cookie regardless of any information on the bowls. In other words, this is the probability of getting a plain cookie from each of the bowls. It is computed as the sum of the probability of getting a plain cookie from a bowl multiplied by the probability of selecting this bowl. We know from the problem statement that the probability of getting a plain cookie from bowl #1 is 0.75, and the probability of getting one from bowl #2 is 0.5, and since Fred is treating both bowls equally the probability of selecting any one of them is 0.5. Thus, the probability of getting a plain cookie overall is 0.75×0.5 + 0.5×0.5 = 0.625.

Pr(B|A), or the probability of getting a plain cookie given that Fred has selected bowl #1. From the problem statement, we know this is 0.75, since 30 out of 40 cookies in bowl #1 are plain.

Given all this information, we can compute the probability of Fred having selected bowl #1 given that he got a plain cookie, as such: (see calculation above)

As we expected, it is more than half.

This type of calculation is routinely employed in statistical evaluation of data particularly in the medical field. To win the Big Prize, explain if you think it is valid or invalid.


rainswept said...

It's valid, but in the non-baking world we seldom know how many cookies are in our bowls prior to testing.

What the Bleep Do We Know = Zampano's Notes
the_Doctor = Johnny Truant


CaptainOrange said...

It seems a great big hole to me. But, how could such a large door be kept secret from everybody outside, apart from the dragon?

Even Bilbo knows that if he has two beautiful seed-cakes and some scones coming out of the oven and a pot of tea ready on the hearth that there is a one hundred percent probability that he has never seen any of these fucking dwarves before in his life.

I gotta go with 0.5 on the bowls. You pick one or the other, what comes next comes next.

mover/shaker/image/maker said...

Hey Dave - first time to the blog. Not sure how the tool is used in med. but my guess is as an aid in diagnosis. In my mind, the difficulty of applying this to bodies versus bowls would come not in the imposissibility of perfect knowledge of the patient body (Bayesian statistics are used for precisely this reason, no?) but of the psychological stickiness associated with the application of the theroem. More specifically, it would seem Bayes' Theorem is meant to adjust preconceived and general knowledge of a disease using specific knowledge from the clinical visit. But here's the rub: the calculation of a prior probability is likely highly subjective. A clinician's impression of the severity of the disease and the prevalence of the disease in popular culture might influence assessment and practice for instance.