# When tossing a coin, what's the likelihood of coming up heads when it came up 999 tails consecutively? [duplicate]

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 KunalanApr 28 '14 at 21:32

What is the definition you have learned for "independent random variables"? – Christopher A. Wong Apr 28 '14 at 20:42
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 '14 at 20:46
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 '14 at 20:47
This question is one duplicate, though I expect there are others. – ShreevatsaR Apr 28 '14 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 '14 at 20:53

Assuming you have a fair coin, then yes, it is $\frac{1}{2}$. Thinking otherwise is known as the gambler's fallacy.
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 '14 at 20:56