Let’s say you have a fair coin and were somehow able to flip it billions of times and recorded the results. Examining the results, you see that the largest amount of times it lands on heads or tails consecutively is 30 times. Now for example, if you were gambling on the results (heads = lose, tails = win) and you witnessed a streak of 29 heads, why wouldn’t you start betting on tails, expecting it to come soon? I mean I know the odds of either heads or tails is 50:50, but wouldn’t it be more logical to expect tails after 29 consecutive heads given that all the data suggests a consecutive streak of either heads or tails hasn’t ever been longer than 30?
In: Mathematics
Coins don’t have a memory. They don’t know that they just flipped heads a bunch of times in a row, and therefore it’s time for them to flip tails.
Because of this, every time you flip the coin, the chance of it being heads or tails is 50/50. It doesn’t matter how many times you just flipped heads or tails in a row (assuming it is a fair coin).
If you just flipped heads 29 times in a row, the odds of getting heads again for the 30th flip is 50%.
You’re saying, “What if I run a simulation and flip the coin billions of times in a row, and the longest streak was 30 times in a row?”
To answer this, imagine running a longer simulation. You flip the coin trillions of times in a row. Now you will have a thousand times among these flips where you got heads or tails 29 times in a row. You can look at all of these times, and you will find that 50% of the time the next flip was heads, and 50% the next flip was tails.
Basically, streaks in the past (which is the only place where streaks can be) tell you nothing about what’s going to happen in the future, and you should ignore them.
> but wouldn’t it be more logical to expect tails after 29 consecutive heads
The probability of a coin *having shown 29 consecutive heads* **after** you happen to have recorded 29 consecutive head throws is **1**. You’ve already recorded it. You know it happened. There is no more chance involved here. All that remains is 1 coin flip, that – as you correctly stated – has a 0.5 probability for each turnout.
The gambler’s fallacy is just attributing new odds, or believing a scenario will be more/less likely, based on a perceived pattern of random events.
In this case the pattern, a coin falling heads/tails 30 times consecutively, is just something you perceive as the observer, whereas the odds are “set”.
If you flipped a fair coin 100 times, and it came up heads 99 times, the 100th flip would still have the same odds (50:50)
If I ask you to put money on the result, you will convince yourself that it MUST be heads since the previous 99 flips were heads. Now, that may be a fine bet, but the odds the coin will fall heads/tails on the 100th flip has nothing to do with the past 99.
I hope this helps. Lmk if you want a non-coin analogy (although coin analogies appear to be the norm)
Gamblers fallacy is exactly the opposite of the strategy you have described.
Gamblers fallacy: I have seen 29 tails so I’m bound for a head soon
Good reasoning: it is extremely unlikely to get tails 29 times in a row. The more likely possibility is that the coin is biased towards tails, and I should guess tails.
Best reasoning: However, in your question, you have made the assumption that the coin is fair. In this case, you MUST attribute the 29 tails in a row to chance, and because it is a fair coin, on the next flip, there is an even chance of having either result.
Latest Answers