# A fair coin is flipped 2k times. What is the probability that it comes up tails more often than it comes up heads? [duplicate]

I'm studying for a probability exam and came across this question. I watched the video solution to it but I don't really understand it. I was hoping someone could explain this problem to me. Are there different ways to go about this?

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## marked as duplicate by amWhy, Hakim, 91500, egreg, Davide GiraudoMay 21 '14 at 12:10

Hint:

The probability that an equal number of tails and heads appear is $\large{{2k \choose k} \frac{1}{2^{2k}}}$

The two remaining outcomes (that there are more heads than tails or more tails than heads) are equally likely.

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It seems likely that the "2k" in the question refers to a general even number $2k$ rather than "2000". – Chris Taylor May 21 '14 at 10:16
@ChrisTaylor: Well, just replace $000$ with $k$, then. ;-) – Ilmari Karonen May 21 '14 at 11:13
When the question was first posted, I believe OP used "a coin is flipped 2000 times." Also, holy cow, why did this question / my answer get so much attention? Wouldn't think that this answer would be my highest rated non-CW answer.. – Michael Tong May 22 '14 at 3:51

Hint:

1. Fair coin $\implies$ Probability of tails occurring more $=$ probability of heads occurring more $= p$, say.

2. Probability of exactly equal number of heads and tails $=1-2p$. Can you find this one?

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@Knickerless-Noggins What on Earth are you talking about? – Did May 21 '14 at 8:47
@Knickerless-Noggins In any particular set of $2k$ throws with a fair coin, there are three mutually exclusive and collectively exhaustive possibilities - (1) exactly $k$ heads (and tails) occur (2) more than $k$ heads occur or (3) less than $k$ heads occur. Why do you think possibilities (2) or (3) have zero probability? – Macavity May 21 '14 at 8:50
@Knickerless-Noggins Think of only one throw. What happens? Does this contradict the probability of equal heads and tails? You may need to brush up on reading or have a chat with someone on this, not much point discussing here... – Macavity May 21 '14 at 10:43
@Knickerless-Noggins: that the coin is fair just means that both sides have an equal chance every single throw. That doesn't mean that 10 throws will always result in 5/5 at all. Try it out for yourself. – RemcoGerlich May 21 '14 at 11:37
@Knickerless-Noggins The probability that a fair coin produces as many heads than tails in $2n$ throws actually goes to zero when $n\to\infty$. Already when $n=5$ ($10$ throws), this probability is less than $25\%$. – Did May 22 '14 at 7:26