Here is the Q&A from Marilyn vos Savant’s Parade column yesterday (I am paraphrasing)
I think her answer is not very good. You will never know one way or the other, you will just “probably” know the answer.
On the other hand, I think there is no solution to the problem as stated.
Any better ideas?
Well, there is a subset of cases that you will know the answer for sure, and that subset is limited to cases that you do not have the gene, as specified.
However, there is a subset of cases where you do not have the gene that you are not told anything. Ergo, there are times when you might be okay and don’t know.
Anything you do to increase the probability of being told if you do not have the gene increases the likelihood that if you are not told that you do have the gene.
So, either you want ambiguity, and have to pay the price of having ambiguity, or you want the answer and have to pay the price of getting the answer. The problem is attempting to only have ambiguity in all cases where you have the gene, and always get the answer if you do not have the gene. There is a fundamental logic problem with that proposition.
Who wouldn’t want to know… That’s what I want to know
Edit:
How’s this for a resolution. (note, this may run into problems dealing with a doctor’s own code of ethics)
Take the test.
Doc tells you if you don’t have the gene.
If doc tells you you have the gene, he says so, but says there’s a 50% chance he’s lying.
Well, by design, you should never know that you actually have the gene. And, in the given setup (supposing the doctor keeps flipping coins forever), in the case where you don’t have the gene, you will, with probability 1, come to know this for sure.
I think that’s pretty much the most a solution can do, given the constraints that you have enough negative introspection to know when you don’t know something [e.g., if the doctor doesn’t tell you on day 1 that you lack the gene, you are aware of the fact that you haven’t been told on day 1 that you lack the gene, and thus can deduce whatever follows from that], and, in the case where you lack the gene, should almost certainly eventually become sure of this, while in the case where you have it, you should never become sure of this.
Put another way: the coin is just an implementation detail. The answer being provided is essentially “If the doctor finds out you have the gene, he never tells you anything. If the doctor finds out you lack the gene, he picks an interval of time by some method which is capable of producing arbitrarily long ones, waits that long, and then tells you that you lack the gene.”
(The only improvement we could even potentially put on this is shortening the delay to learn that you lack the gene; but any finite upper bound put on that wait time would mean that you eventually learn that you have the gene, just by waiting that long and observing if you fail to be told that you lack the gene.)
How about this:
If you don’t have the gene, the doctor tells you.
If you do have the gene, the doctor tells you that the test was inconclusive and will not work on you. (This is, of course, a lie).
Now, this obviously doesn’t work if the person asking the question knows that the doctor will lie on a positive result. But the doctor could take the original question and apply this as a solution that will fit the criteria asked for.
If you don’t have the gene, the doctor tells you. If you have the gene, the doctor pulls out a gun and shoots you in the head.
As I read the problem, there’s no requirement that you don’t end up believing anything false in the case where you do have the gene (should be allele, but that’s nitpicking). So it’s perfectly fine if the doctor performs the test and then tells you that you don’t have the gene regardless of the outcome.
But, for a brief second, you’d know! Oh, what a terrible weight such knowledge would be to carry…
But, as I read the problem, the patient is meant to be aware of the method the doctor uses to give his responses. So they don’t necessarily just listen to the doctor blindly; they take their knowledge of the method the doctor uses, combine it with the words he speaks, and come to whatever valid conclusions they can. Thus, if the doctor says the same words regardless, the patient will never learn anything, even in the case where they lack the gene.
That’s weird. I read the OP and was about to post exactly this - almost word for word.
Exactly. The coin toss is for show.
If I were a doctor I would suggest this patient seek further counseling from a genetic counselor, a family counselor, a psychologist and/or more than one of the above, so as better to come to terms with the possibility of inheriting the disorder.
The solution offered is wrong. Each flip gets you closer to getting a true answer, as the chance not to flip tails gets smaller. After only 3 times, it’s 12.5% This means a 12.5%*50%=6.25% chance you don’t have it but won’t be told, a 87.5%*50%=43.75% chance you will be told you won’t have it, and, of course, a 50% chance you do have it. Looking at this another way, if you aren’t told, you have a 6.25/56.25= 11.11% chance of not having it, but a 50/56.25=88.89% chance of having it.
Compare that with just one flip, which has a 50% chance of hitting tails, meaning a 25% chance of being told, a 25% chance of not having and not being told and not having it, and a 50% chance of having it. Not being told, that means you have 25/75=33% chance of not having it, and a 50/75=67% chance of having it.
Here’s a chart with all the numbers. (Hope it makes sense):
%NOT TOLD WHEN NOT TOLD
TOSSES %TAILS %TOLD &HAVE &NOT HAVE %HAVE %NOT HAVE
1 50.00% 25.00% 50.00% 25.00% 66.67% 33.33%
2 75.00% 37.50% 50.00% 12.50% 80.00% 20.00%
3 87.50% 43.75% 50.00% 6.25% 88.89% 11.11%
4 93.75% 46.88% 50.00% 3.13% 94.12% 5.88%
5 96.88% 48.44% 50.00% 1.56% 96.97% 3.03%
6 98.44% 49.22% 50.00% 0.78% 98.46% 1.54%
7 99.22% 49.61% 50.00% 0.39% 99.22% 0.78%
8 99.61% 49.80% 50.00% 0.20% 99.61% 0.39%
9 99.80% 49.90% 50.00% 0.10% 99.81% 0.19%
10 99.90% 49.95% 50.00% 0.05% 99.90% 0.10%
11 99.95% 49.98% 50.00% 0.02% 99.95% 0.05%
12 99.98% 49.99% 50.00% 0.01% 99.98% 0.02%
13 99.99% 49.99% 50.00% 0.01% 99.99% 0.01%
14 99.99% 50.00% 50.00% .003% 99.99% 0.01%
15 100.00% 50.00% 50.00% .002% 100.00% .003%
By the time you get to 15 tosses, we’re down to less than a 1-in-30,000 chance that you don’t have the disease if the doctor doesn’t say anything. And it will keep getting smaller, until there’s no practical difference between the two answers.
So why not just pick a low number? Remember, you start off with a 1 in 3 (33%) chance of not having it. That means that, even not being told once tells you more information than you had before, and thus fails the question.
Every solution has this in the limit: if, when you lack the gene, you approach probability 1 of eventually knowing this [i.e., P(I know I lack the gene by time t | I lack the gene) approaches 1 as t approaches infinity)], then, correspondingly, P(I fail to know I lack the gene by time t | I lack the gene) approaches 0 as t approaches infinity, and thus, as P(I fail to know I lack the gene by time t | I have the gene) = 1, we have that P(I lack the gene | I fail to know I lack the gene by time t) approaches 1 by typical application of Bayes’ rule.
So if, when you lack the gene, you can be almost certain of this in the limit, then the same holds of the case where you have the gene.
Accordingly, the best we can do is worry about the distinction between what you know to have a high probability of being true and what you know to be guaranteed to be true: make it so that, when you lack the gene, you’ll arrive at a state where you have definite knowledge that you lack the gene, and when you have the gene, you’ll never arrive at a state where you have definite knowledge that you have the gene. And this, the proffered solution accomplishes.
The specific probability distribution used to accomplish it doesn’t matter, as they’re all the same in the limit; indeed, probability itself doesn’t really matter. Probability is, despite this post, a red herring.
All that matters, like mentioned above, is that you use a method where, if you have the gene, the doctor never says anything, and if you lack the gene, the doctor somehow picks a potentially arbitrarily long length of time to wait before announcing that you lack the gene, never saying anything before then.
I knew all solutions would eventually fail. But this one fails rather quickly. It takes very little time to flip a coin 15 times, and by then you’ve already established that you are more than 99.99% sure.
I’m not sure at what probability that the statistics become meaningless. The formula for how unsure you are is (1/2^n)/(1-1/2^n), where n is the number of flips. I know you can mathematically say you’ll never be sure, but you can be as sure as you are about anything you consider absolutely certain. At 21 flips, you’re more sure than you’d be that the sun would rise if you’d watched it happen for 5000 years.
It fails rather slowly, if you only flip the coin once a year. 
Like I said, the coin is an implementation detail. If you want to analyze the situation probabilistically, then pick whatever probability distribution on wait-times you like; distributions will have longer average wait-times and P(I have the gene | I haven’t been told I lack the gene by time t) approaching 1 more slowly. But they’re all basically the same, on some reparametrization of time…
How is this for a solution:
Er, what a spectacular mess of editing: the middle sentence should have originally had the words now placed in bold.