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Consider that you're tossing a coin $1000$ times. First $999$ times, it came up heads. Is it $50\%$ chance that the coin tails after $1000^{th}$ trial? Are these discrete events?

Since the mean of these experiments is $500H$ $500T$, each heads increases the probablility of tails, doesn't it?

Edit: Let me ask the question that way: If I toss the coin 10 times, $5H$ and $5T$ is the mean.

If $TTT$ comes up, then probability of $H$ would be $\frac{5}{7}$, would it not?

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marked as duplicate by ShreevatsaR, MJD, Davide Giraudo, Sami Ben Romdhane, Sujaan Kunalan Apr 28 at 21:32

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What is the definition you have learned for "independent random variables"? –  Christopher A. Wong Apr 28 at 20:42
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Do you know that this is a fair coin, or are you trying to determine whether it is fair? If this happened in real life I would want to examine the coin to make sure it did not have heads on both sides. –  David K Apr 28 at 20:46
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The coin has no memory. See gambler's fallacy. This question has been asked and answered many times on this site before. –  ShreevatsaR Apr 28 at 20:47
    
This question is one duplicate, though I expect there are others. –  ShreevatsaR Apr 28 at 20:48
    
By the way, if you see TTT, the expected number of heads and tails among the remaining $7$ is $7/2$ each, so the expected total number of heads and tails (among the ten tosses) would now be $7/2 = 3.5$ and $3 + 7/2 = 6.5$ respectively, no longer $5$ and $5$ (that's only the expected value before the results of any tosses). –  ShreevatsaR Apr 28 at 20:53

1 Answer 1

up vote 3 down vote accepted

Assuming you have a fair coin, then yes, it is $\frac{1}{2}$. Thinking otherwise is known as the gambler's fallacy.

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And if you don't assume you have a fair coin, then the fact it comes up heads $999$ times and has never been observed to come up tails is evidence that the coin is not fair, and that heads is more likely than tails (not less likely) on the next flip. –  David K Apr 28 at 20:56
    
If that coin came up tails 999 times in a row, the assumption that you have a fair coin is just ever so slightly unrealistic. –  gnasher729 Apr 28 at 21:19
    
@DavidK Agreed. I never understood why anybody would think a huge run of heads would make tails more likely. If tails were "due", it probably should've happened before 999 heads came up. The reverse gambler's fallacy, while also incorrect, at least makes some sense. Plus it actually works with unfair coins. –  RandomUser Apr 28 at 21:27
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Right. I think the most reasonable interpretation of the original question is that the coin is fair (even though the word "fair" was not used), hence your answer is the correct one. But I find it amusing that in the most plausible real-life alternative interpretation of the problem, the gambler's fallacy is still wrong. –  David K Apr 28 at 21:51

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