Flipping a tail is a 1/2 chance, but flipping 6 tails in a row is a 1/64, so if after flipping 5 tails, why is it incorrect to say that your chance of flipping another tail is now lower, like you’re “bound” to get a head? I know this is the gambler’s fallacy, but why is it a fallacy? I get that each coin flip is independent, but it feels right (as fallacies often do) that in consecutive flips the previous events matter? Please, help me see it in a different way.
In: Mathematics
Let us pretend that a machine flipped a coin 6 times and it recorded each result.
HTHTHT would be heads-tails-head-tails-heads-tails
HHTTHT would be heads-heads-tails-tails-heads-tails
Now this machine does this 6 flip and record exercise one million times, and you have access to that table.
If you go look for how many of the entries are TTTTTT (six tails) you will find that it is 1/64 of the total number of recorded entries.
Next you go look for every entry that starts with five tails, to TTTTT* where the * is either H or T. If you put all of those entries together you will find that all of them are 2/64 of the total of all recorded entries. And if you look at them the number of TTTTTT entries and TTTTTH entries are the same.
From an emotional point of view, it does feel like the heads is “due” to come up. But if it helps think of it this way “what are the chances of me flipping 6 tails in a row?” is a different question than “what is the chance of me flipping a tail, given that there were 5 flips before it?”
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