How is it that when you flip a coin 10 times, the likely hood that it’ll land on heads 10 times in a row is extremely small but the likely hood that it’ll land on heads is 50/50 if it already landed on heads 9 times? I get that it’s a closed system and its roughly 50/50 for every coin flip but my brain is just telling me that it should be a higher chance that it would land on tails instead of heads. How does this work?
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Each toss is its own event. The past coin tosses have no effect on the outcome of the current coin toss. 9 heads in a row would be a 1 in 512 occurrence. 10 heads in a row would be a 1 in 1024 occurrence. Likewise, a person playing blackjack, whom lost 10 hands in a row, is not due a winning hand.
From the mathematical point of view:
P(10 times heads) = 0.5 * 0.5 * 0.5 * 0.5 * 0.5 * 0.5 * 0.5 * 0.5=0,0009765625
P(1 time head after 9 times head)=0.5
Probability multiplies with every tossed coin, if you are interested in only one outcome after 9 other its only this outcome that matters.
So the chances of flipping on heads three times in a row is fairly small, 12.5% chance, but no matter what the chances of any coin landing on heads is 50%
We instinctively know it’s unlikely for it to land heads 10 times in a row, so when it gets to 9 we think it must not land on heads since that would result in something so unlikely, this is of course a fallacy and it’s always 50%.
It sounds like you fully understand the gambler’s fallacy because you’ve just explained it and you’re experiencing it.
Are you then asking how does the brain think that it has a high chance of landing on tails?
After 9 tosses, the chance you have 9 heads is ~1/500.
On the tenth toss – you still have a 50% chance of heads or tails.
You may now have ten heads which is pretty rare, about ~1/1000.
Or you may have 9 heads and a tail – equally as rare, ~1/1000.
In fact any sequence of 10 heads/tails is equally as rare.
Comparing chance of ten heads to not-ten-heads: 1/1000 vs 999/1000
I heard a story from a public speaker about an event that he went to. There were exactly 1024 people at the event. One of the organizers asked another if he wanted to bet him that he could find one person to correctly predict a coin toss 10 times in a row. The other person did not believe this was possible and took the bet.
The organizer smiled and told everyone of the attendees to pair up. Then each one of them had to choose heads or tails. The one who chose correctly would move on, the other would not. With the number of people there it was exactly 10 rounds. One person had called it correctly 10 times.
Now, I’m not sure if this is real but it illustrates the point. Calling heads or tails 10 times in a row seems hard but if you take that context away it’s just a single guess repeated 10 times. Another way to think about it is what if they were spread out? Instead of “10 times in a row” you just do it once a day and record the result. Does it seem that hard? I think proximity of the choices has a lot to do with it as well.
That’s why it’s a fallacy- our EXPECTATIONS do not match REALITY.
And when you include an emotional response or stress, it becomes much harder for you to stop and look at it rationally. So the problem only exists in our minds, the coin doesn’t care. The coin doesn’t remember what the result was the last nine times you flipped it.
The whole problem humans have with the gambler’s fallacy is that although we have a memory, the coin does not. So you just flipped a coin 9 times in a row and it was heads every time? WE remember that unusual occurrence, for the coin it does not make a difference at all what it landed on previously when you flip it now. When you do a flip it is always like starting a completely new series. Unless of course the coin isn’t balanced.
The odds of 10 coin flips in a row being heads is 0.5^(10), or 0.09765625%. Incredibly low. So the gambler thinks, well there’s already been 9 heads so far, surely it must be time for a tails. The odds of 10 heads is so small!
You know what the odds of 9 heads and then a tails is? 0.5^(10) or 0.09765625%. Yep, exactly the same. The coin doesn’t care that the last 9 flips were heads. The next one is still either heads or tails, 50/50.
It’s pure psychology. Humans are pattern recognition machines. We’ve assigned special meaning to 10 heads in a row, thinking it’s unlikely to occur by chance, assuming a fair game. You can’t get it out of your head because your head is wired to give such outcomes significance. You think 10 heads is special. But you don’t care about 9 heads and a tail. Or 1 tail then 9 heads. Or if the 5th one was a tail and the rest were heads. Those are just normal random occurrences. But statistically all these outcomes have exactly the same chance.
I struggled with this for a while too until some one explained it this way:
Image there are two different coin challenges.
* Challenge 1 is to get a single “heads” flip.
* Challenge 2 is to get ***ten*** “heads” flips in a row.
It’s easy to see why for the first challenge your chance at success is 50/50. You have exactly one opportunity to succeed and one opportunity to fail.
Now look at the second challenge. Ten “heads” in a row. Again you have only one opportunity to succeed (getting a heads for every flip), but now you have 10 opportunities to fail. You could get 7 heads in a row, but if you get tails on the 8th flip, then the whole thing fails.
When you think about getting heads 10 times in a row, it’s the second challenge. But when you think about getting heads just one time (even if you’ve gotten heads the last 9 times) it’s still just the first challenge.
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