Assuming a fair coin, 50%. On the other hand, that coin only has a .00098 probability of doing that by random chance. Since that's well below .05, I'd be perfectly justified in concluding that I'm flipping a double-headed coin. Unless I have some other data to the contrary, of course.
I see why you're asking me, though. My example was poorly phrased. The first three coin flips do not have any affect on the probabilities for the last three. However, they can dramatically affect the probabilities for the total final distribution, since they're counted in that distribution. To illustrate, if I flip six coins, I have a .334 probability of getting at least four tails. However, if I flip the first three and they all come up tails, I now have a .875 probability of getting at least four tails in the final tally, because three of my four needed tails are already locked in. The Gambler's Fallacy "predicts" (note the quotation marks) exactly the opposite effect. That even though the first three have already come up tails, the probability of getting a fourth tails has not changed. The Gambler's Fallacy suggests, wrongly, that the probability would remain .334, because somehow the remaining three coin flips would be more strongly inclined to turn up heads.
tl;dr: Reality predicts that the first three coin flips affect the total distribution, but specifically not the last three coin flips. The Gambler's Fallacy erroneously predicts the opposite: that the first three coin flips affect the last three coin flips, and thereby do not affect the total distribution. That's what I was trying to illustrate earlier.
@Jarrakul If you flip a coin 10 times and it somehow turns up heads every time, what are the odds the next flip you will get tails?
Well, the odds for the NEXT flip are ALWAYS 50/50. But I think people get confused because in the big picture, the odds of ten or eleven straight "heads" resulted are very slim. But you need to look at each event as a unique occurrence and not be confused by the big picture.
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I see why you're asking me, though. My example was poorly phrased. The first three coin flips do not have any affect on the probabilities for the last three. However, they can dramatically affect the probabilities for the total final distribution, since they're counted in that distribution. To illustrate, if I flip six coins, I have a .334 probability of getting at least four tails. However, if I flip the first three and they all come up tails, I now have a .875 probability of getting at least four tails in the final tally, because three of my four needed tails are already locked in. The Gambler's Fallacy "predicts" (note the quotation marks) exactly the opposite effect. That even though the first three have already come up tails, the probability of getting a fourth tails has not changed. The Gambler's Fallacy suggests, wrongly, that the probability would remain .334, because somehow the remaining three coin flips would be more strongly inclined to turn up heads.
tl;dr: Reality predicts that the first three coin flips affect the total distribution, but specifically not the last three coin flips. The Gambler's Fallacy erroneously predicts the opposite: that the first three coin flips affect the last three coin flips, and thereby do not affect the total distribution. That's what I was trying to illustrate earlier.
My apologies for being unclear.