Gambler’s Fallacy and the Regression to the Mean

Well, a "post-hoc description" of empirical observations is how we make predictions. We gather cases (and their probabilities) post hoc, and predict future results based on that.

Why shouldn't it be applied in this case, especially when here not only we can calculate the probabilities post-hoc, but even use probability to pre-calculate those post-hoc probabilities for every series of streaks?

>The only way you can reasonably expect the past streak to have an impact on the next spin

Regression to the mean is a reason to expect past streaks to have an impact on future spins. Doesn't have to be some force that specifically affects the individual next spin, but it's a probability that affects (and moves toward the mean) the outcome of next spins.

Intuitively speaking, why shouldn't the gambler use it?

If they shouldn't, "because spins are independent" seems too hollow as an answer, because we have independent spins but also a statistical property for their aggregation.

I guess we could test it empirically: in a fair sequence of coin tosses, would a gambler always betting on tails after seeing N (for e.g. N > 4) consequtive heads have an advantage, over one always betting randomly on the same N+1 toss?

But regardless of that, my point is that intuitively is quite justified, and the explanations why it's not seem more hand-wavy.

>Try this on: What is the "mean" red-black on the roulette wheel? There isn't one.

Well, (ignoring non red-black case), there's not a mean value, but for the distribution there's an expected long term tendency of a balanced red and black outcomes.

>... And editing your comment to refer to groups of throws is disingenuous. It would have been more honest to modify your case in a reply.

I didn't modify my case, I expressed it more clearly. What exactly was disingenuous? I more often than not post a quick comment (HN habbit, as sometimes old edit forms expire), and then re-read it and quickly improve it as I re-read it, find typos, better wording, etc. Didn't even think anybody would read it and answer in the time it took me to refine it.

And, how about it, you can always answer to my final, improved, formulation. Does it have any historical interest if you answer to a worse worded case?

No.

If the host chooses a door randomly and it doesn’t happen to contain the prize, the outcome is exactly the same as if the host does know where the prize is. Look up “conditional probability” for how to calculate this. The host’s knowledge has no effect other than to prevent the awkward outcome in which the host opens the door with the prize.

Even if the host did open the door with the prize on occasion, the outcomes would still be the same if the rule was that the whole game starts over if the host accidentally reveals the prize. (Well, the game would take longer to finish, but that has no effect on the probability of winning.)

Consider the first one hundred spins breaking 60 red, 40 black. That is a little odd, but not outrageously so. The ratio is 60/40 = 1.5

Suppose that the next one hundred spins break 50/50. There are no surplus blacks to push the ratio towards 50/50. So what is your intuition? It seems clear enough that the totals are 110 red, 90 black, for a ratio 110/90 = 1.22...

Having imagined the next hundred spins vividly, we are comfortable saying "Unless the surplus reds keep coming, the ratio is going to drift back towards 50/50 all by itself.". There is a striking clash between different "things we might say" seeming sensible, depending on the vividness and detail with which we imagine the situation.

Well, kind of. There are 2^6 patterns with 6 tosses, and only a few of them have some recognizable pattern (are compressible). HHHHHH is the worst of them. It might be "equally likely" to any other 6-toss pattern seen from a fair coin, but it's also a great example to make one suspect a totally rigged (same sided H) coin.

Now, on average a fair coin would have roughly balanced H and T. So the (much more numerous and thus probable) series where H and T are mixed (e.g. HTTHHT..., HTHTHH..., etc...), are more probable than a large streak of H.

>The probability that there will be one T somewhere in the sequence is a lot higher than the probability that there will be no T. That's only because there are a lot of somewheres for the T to be, but only one way for there to be no T.

A, my sentiments exactly!

>But once you have already seen 99 H, you still have precisely zero information about the next fair flip, and there's no way around it.

That's my argument: you don't have zero information.

You know that the next fair flip is random - sure.

But you also have the information that large series of tosses end up balanced in the long run, which is a kind of push (given fairness) towards making a T after tons of Hs increasingly likely.

Why should this be discarded?

Even if you could, you already know that you're losing, unless he doesn't show you what's behind it.

If the game is still going, you know that he picked an empty door regardless of whether he knew he was doing it