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This is a weird problem that popped into my head: given a fair coin, how many flips is required to guarantee heads?

If I get a tails, then another tails, and another etc., the chance of getting a heads increases every time. But there is still a small chance that I will get another tails.

This seems to imply that there is no finite number of flips to guarantee a heads. Does this mean that infinity is the correct answer (although infinity isn't a number as far as I understand) or is this question even answerable in the first place?

Ken Duna
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imas145
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    You can theoretically flip an infinite number of tails: there is no magical force in the universe that will guarantee you will ever flip a heads. That being said, the "probability" of flipping an infinite number of tails is $0$ -- this kind of event is called almost surely impossible https://en.wikipedia.org/wiki/Almost_surely#Tossing_a_coin – Szmagpie Jun 11 '16 at 18:08
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    How do you know how many tails came up on the coin, before you started your experiment? How does the coin know? – DJohnM Jun 11 '16 at 19:21
  • Correct: infinity is not a number, and has no business being in computations that don't employ "limits". – Mazura Jun 11 '16 at 21:52
  • After a few billion flips and landings with heads-down, enough material from ridges, etc., on the heads face might get worn off that it would no longer be a fair coin. – user2338816 Jun 12 '16 at 00:06
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    @imas145: you may be interested in looking into the concept of independence in statistics. Assuming a coin is flipped the same way every time, landing on tails doesn't affect the possibility of it landing on heads or on tails in future flips. – Jules Jun 12 '16 at 01:42
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    See also: Gambler's fallacy. – Clive Newstead Jun 12 '16 at 02:04
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    @Mazura: Is infinity a number? –  Jun 12 '16 at 07:09
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    Thank you everyone for correcting my assumption on the increasing chance of getting a heads. – imas145 Jun 12 '16 at 07:53

3 Answers3

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There is no way to guarantee that you will get a heads ever. The chance of getting heads remains a constant 50-50 on each individual flip--flips are said to be independent. It is only in the aggregate of an increasing number of flips that the probability of getting a heads on at least one flip increases. However, while this probability increases monotonically, it never reaches 1.

Yes; it is extremely unlikely that you will get 5 million tails in a row, but it is entirely possible. You can answer a similar question if you are willing to set a tolerance. I.e. if you wanted 95% confidence that a heads will appear, then you want the probability that $N$ flips in a row are tails to be less than 5%.

The probability that $N$ flips in a row are tails is $(0.5)^N$. Computing this for different values of $N$:

\begin{array}{ll} N & 0.5^ N \\ 1 & 0.5 \\ 2 & 0.25 \\ 3 & 0.125 \\ 4 & 0.0625 \\ 5 & 0.03125 \end{array}

Therefore flipping the coin $5$ times will give you $(100 - 3.125)$% = $96.875$% confidence that a heads will appear at least once.

geometrian
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Ken Duna
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Basically the answer: "infinite" is the correct one. Moreover, you have to be careful in saying that the chance of obtaining a Head increases 'every time'. This statement is false. Each time the chance of obtaining a head is $1/2$. What is true is that the chance of obtaining at least a head in $n$ throws is $1-2^{-n}$. This probability clearly increases with $n$, but you should appreciate the difference with respect to your statement.

guestDiego
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If I get a tails, then another tails, and another etc., the chance of getting a heads increases every time.

no. it's still a fair coin after you throw several tails. it's always 50% odds of heads or tails on the next throw. this is just as superstitious as thinking you're at a hot craps table at the casino.

curi
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