5 ms·

HH is less likely than H (.25 vs .5), but HT is also less likely than H and by the same amount. HH and HT are equally likely. This extends to HHHHHH and HHHHHT,

by psyc 5y ago

HH is less likely than H (.25 vs .5), but HT is also less likely than H and by the same amount. HH and HT are equally likely. This extends to HHHHHH and HHHHHT, and so on. There’s no way to frame the gambler’s fallacy that makes it a better bet. 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. 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. If you must make predictions about the streak as a whole, you have to lock them in before the first flip. Otherwise you piss off both Claude Shannon and Tony Soprano.

coldtea 5y ago
>HH is less likely than H (.25 vs .5), but HT is also less likely than H and by the same amount. HH and HT are equally likely. This extends to HHHHHH and HHHHHT, and so on. 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?
- psyc 5y ago
  > Why should this be discarded? Well, it's a restatement of the Gambler's Fallacy, but I'll taboo using that as an argument because it begs the question (but for reals!) I thought about this for a while, and here's the best I can do for right now. It's like winning the MegaMillions lottery on Saturday, and then winning $1 on a scratch ticket on Sunday. You'd go running around screaming on Saturday, but you wouldn't on Sunday. (Would you?) Because the impossible already happened. Hitting $1 on a scratch-off is never remarkable (unless you're 8), including on the day after you won the MegaMillions. It's just a somewhat anti-climactic coincidence.

by psyc 5y ago

coldtea 5y ago

>HH is less likely than H (.25 vs .5), but HT is also less likely than H and by the same amount. HH and HT are equally likely. This extends to HHHHHH and HHHHHT, and so on. 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?

psyc 5y ago
> Why should this be discarded? Well, it's a restatement of the Gambler's Fallacy, but I'll taboo using that as an argument because it begs the question (but for reals!) I thought about this for a while, and here's the best I can do for right now. It's like winning the MegaMillions lottery on Saturday, and then winning $1 on a scratch ticket on Sunday. You'd go running around screaming on Saturday, but you wouldn't on Sunday. (Would you?) Because the impossible already happened. Hitting $1 on a scratch-off is never remarkable (unless you're 8), including on the day after you won the MegaMillions. It's just a somewhat anti-climactic coincidence.