Comment by ajanuary | Hacker News Reader

Comment by ajanuary | Hacker News Reader

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I did an experiment: I kept flipping two coins [1]. If it came up tails, tails I ignored it. If it came up heads, heads I noted it down in one column. If it came up heads, tails I noted it down in another column. After many flips the ratio was 1:3.

As far as I can tell this is an accurate simulation of the problem. We have two things, coins or children, which have a 50:50 chance of being one value or another. We are told one combination of values isn't the case, so we discount them from the simulation. We then use the frequency of the target combination to approximate its probability. We haven't added any artificial ordering into the problem.

I know it's not well argued logical reasoning, but it supports the articles approach. Can you point out where I tripped up in the simulation?

[1] Of course I didn't actually sit there flipping coins. I wrote a program to do it. (let [sample-size 10000000] (/ (get (frequencies (take sample-size (filter #(not= % [:girl :girl]) (repeatedly #(vector (rand-nth [:boy :girl]) (rand-nth [:boy :girl])))))) [:boy :boy]) sample-size))

Instead of actually programming, I flipped 2 coins this morning and left them on my desk. They are still sitting there. Either HT, TH, HH or TT

I just remembered one of them is tails, so is there is a 1/2 chance that they are both tails now? Nope it's 1/3 chance they are both tails. Either HT, TH, or TT.

Of course that's a much simpler way to simulate the problem, but then you get people trying to argue that HT and TH should be collapsed into a single state.

By using a simulation with repetition it's easier to pose the states as 'TT' and 'not TT' and use frequencies to show 'not TT' occurs more frequently than 'TT'. I find this starts to help convince people that the states shouldn't be collapsed, even if it doesn't explicitly explain why.

>We haven't added any artificial ordering into the problem.

Yes you have! To run the experiment, you have to treat TH and HT as different but you "pretend" not to do so by adding them into the same pile. You're pretending to know that the father tells you which child is the "known boy", but by the statement of the problem, you don't know that. Why is this an error? Because the father must follow a rule - he must give you a boy, and there are two ways for him to do that in BB but you only recognize one. If you go the "table-minus" route, which both the author and your experiment do. But you can't do that without affecting the odds! By telling you they aren't both girls, he lowers the chance that either of them might be. Yes, I'm serious.

Correct formulation of table:

a) BG; b) GB; c) BB and he gives you the first; d) BB and he gives you the second.

Isn't the probability of c and d half that of a and b? If BG is the case, there's 100% chance he's talking about the first child. If BB is the case, there's a 50% chance he's talking about the first one, and a 50% chance he's talking about the second one.

    BG, giving the first - 1/3
    GB, giving the second - 1/3
    BB, giving the first - 1/6
    BB, giving the second - 1/6

[Edit] You're right. Tables can be a bad way of thinking about it, because you have to make sure each option has an equal probability for them to work. This is okay when you have a single variable (like in the original post), but gets tricky when you introduce a new variable (as you did: is the father thinking about the first or second). Once you get past one variable, trees are probably a better but still simple representation.

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