I know it’s true, it’s just something that seems hard to wrap my head around. How is this not a logical paradox?
In: Mathematics
We’re the ones calling “25 heads in a row” something special. If your favorite pattern of choice was “3 heads, 2 tails, 5 heads, 1 tail, 1 heads, 7 tails, 1 head, 2 tails, 3 heads” then THAT would be the one seeming super unlikely.
The coin has no clue about the previous flips. It doesn’t say “oh all right I’ve been heads 24 times now, I’m tired, let’s do a tails for once”. It’s just dumb physics and every flip has the same rules. We’re the ones looking for patterns and being surprised if a pattern emerges. Every particular pattern is just as unlikely as any other particular pattern.
But one specific pattern of choice is of course much less likely than “any random old pattern”, because there are countless such random old patterns and only one “favorite” pattern. We lump the random patterns into one category and we put the special pattern alone in its own category. Then of course, “25 heads” is super unlikely compared to “anything else” because there is so much more of “anything else”.
25 flips being heads is only unlikely because any of those 25 flips have a chance of being tails. After you’ve already flipped 24 heads, it’s no longer true that any of the 25 flips could be tails. Now there’s only one flip left that could be tails. The previous 24 flips are unable to be tails at this point. Your 25th flip can’t cause the already-set-in-stone 1st through 24th flips to change their results.
This assumes a fair coin, of course. After too many flips I would start to question the probability of living in a universe where this exact pattern happened versus living in a universe where the coin is biased and someone is cheating.
My brain won’t fully accept it either (if you really want your brain to hurt, look up the Monty Hall problem) but here’s the way I see it. There are 2 separate math problems here. The first one is “what are the odds of getting heads 25 times in a row” vs “what are the odds of getting heads just this one time”. You could then add 1 more flip and calculate the odds of getting HH. Then add 1 more flip, calculate the odds for HHH, and so forth. You’ll see the odds go way down pretty soon.
I don’t know whether this hurts or helps you, but imagine that you want the results of 25 coin flips to be HHHTTHTHTTHHHTTTHHTHTTHHT. You know the outcome can’t be a 50/50 chance. That’s a very specific pattern. And in fact has the exact same chance of happening as HHHHHHHHHHHHHHHHHHHHHHHHH.
Please let me know if this doesn’t help, so I never suggest it again, lol.
It’s because each flip of the coin isn’t affected by previous flips it’s still a 50/50 chance. The odds of getting heads after 24 tails is the same as getting tails after 24 tails 0.5^25
Simply because you can just ignore the previous outcome. As a coin flip is completely random it will not care about what happened to it before, it will still have the 50% chance of being a head or a tail.
Any specific combination of 25 flips would have the same odds. Singling out them all being the same only has psychological significance, not statistical significance.
Past results are not an indicator of future performance. The coin doesn’t know what happened.
Considering both gambler’s fallacy and logical paradox are outside the understanding of most 5 year-olds, I’ll use similar level terms to what were in the question. The gambler’s fallacy (and indeed most fallacies) exist because they seem like they should be sound based off of what society assumes. Or, based off of what society assumes to be true. The fact that you think it is a logical paradox shows that you are susceptible to this fallacy yourself and should probably avoid gambling. Not meant as an insult, everyone has some logical fallacy they believe in, it’s human nature.
The coinflip one you brought up only makes sense if the outcome of previous coin flips somehow change its balance. But in this case, you would have to know what it landed on every time it was dropped, flipped, or jangled in its life. This is a false scenario though so doesn’t matter, other than to show the absurdity of past flips altering the probability of the next one.
If, before a single flip, you were to bet on 25 flips in a row being heads, yes, those are some long odds. If, after 24 heads you stop and ask what the odds are the next one is heads, its still 50/50 because, well, past flips physically don’t change the coin so individual flips are still 50/50. It would be the same as saying “I’m going to flip this coin 25 times. What are the odds the last one is heads?” It’s 50% chance, from the first flip, that the coin will land on heads at number 25.
TL;DR Individual flips are always 50/50 and past events and perceived future events don’t change that. The fact that this seems illogical to some people is exactly why the fallacy exists in the first place.
If you flip a coin three times, the chance of getting three heads in a row is 1/8, which makes sense because there are eight possible outcomes and they’re all equally likely.
The eight possible outcomes are
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
If after the second flip you’ve gotten two heads, then you eliminate all but the outcomes that had a tails in one of the first two spots.
HHH
HHT
~~HTH
HTT
THH
THT
TTH
TTT~~
There are only two possible outcomes remaining and they’re still equally probable, so there’s a 50% chance of getting HHH, and 50% chance of getting HHT.
The same logic works for 25 coin flips. All of the millions of possible outcomes with a tails in one of the first 24 flips have already been eliminated, leaving only two possible outcomes left.
If you flip a coin that you know is fair, then you would correctly expect even odds for heads and tails. You’d expect so even if you use a coin that has been flipped hundreds of times before. For a fair coin, the relation between past and future flips is entirely in your mind.
The probability of flipping 25 heads in a row is very low, but it’s only half as much as flipping 24 heads in a row, so the odds of flipping the 25th heads after already getting 24 is the same as getting heads once.
Latest Answers