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I recently found a half dollar coin in my basement. It's strange but I think when you flip it, that heads is a big favorite to land face up! I really could have sworn I heard that somewhere before; am I wrong or is the half dollar weighted for heads?

It's funny, I actually said to my friend yesterday, "Here look at this coin, I think it's weighted for heads, like 3 out of 4 times it lands heads." Well I flipped it 4 times and that's exactly what happened.

In my opinion it's because the coin is large and the face seems to be pretty heavy; I'm curious to know if anyone can derive the exact chance of a half dollar landing heads, or to convince me it's not weighted.

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I have convinced myself that it's not weighted: I flipped a half dollar coin four times and it landed on heads twice. –  Greg P May 4 '11 at 1:13
    
Mine is from 1974, I just flipped a bunch of times and I'm telling you this thing is top heavy, u can just feel it when you hold the coin. –  Matt May 4 '11 at 1:18
    
Oh, ok. Mine is...not from 1974. –  Greg P May 4 '11 at 1:25
    
Lets say that the coin was completly uniform, yet it kept coming up heads more often, would you consider that coin to be weighted? –  Arjang May 5 '11 at 8:54
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Please flip it 1000 times, and then ask the question. You will probably see a bias if it is greater than 4%. –  Ron Maimon Sep 3 '11 at 4:00
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3 Answers

up vote 10 down vote accepted

The flipping of a half dollar has been investigated (theoretically and experimentally) by the great Persi Diaconis along with Susan Holmes, and Richard Montgomery: see

http://www-stat.stanford.edu/~cgates/PERSI/papers/headswithJ.pdf

It is rather impressive. They conclude that "vigorously-flipped coins are biased to come up the same way they started" in that a coin has about a 51% chance of landing on the side that started up when the coin was flipped. They call this a "dynamical bias." It has nothing to do with the asymmetry of the coin (which they take into account - they even give the principle moments of inertia of an American half dollar). If the initial condition (heads up or tails up) is chosen randomly (and fairly!), then according to Diaconis et al. the result is essentially a fair coin toss. This of course raises the question of how this initial state should be generated in practice. But in any case this dynamical bias, in practice, favors heads and tails equally.

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"how this initial state should be generated in practice" - simple, just toss a coin! –  Martin Beckett May 4 '11 at 20:05
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This is irrelevant for the question at hand, because the question is asking about a particular coin, a silver dollar, and this paper is not about this specific coin. –  Ron Maimon Sep 3 '11 at 4:01
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Measuring the probability of heads and it being weighted are mutually exclusive up to a point. To measure the probability it needs to be flipped statistically, to measure if it's weighted then need find its centre of mass.

There are at least two types of weighted that needs to be considered:

Physically : you can measure if one side is heavier than other. But this does not neccerily imply that you would get one side coming up more than other. (This is the physics forum answer)

Mathematically : Toss the coin and see what happens statiscally, the coin might be weighted (physically or not) but if one side turns up more than other then yes it mathematically weighted : http://www.vosesoftware.com/ModelRiskHelp/Analysing_and_using_data/Bayesian/Bayesian_analysis_example_identifying_a_weighted_coin.htm

Need to correlate the both to see at what point one type of weighted can be inferred from the other.

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To determine EXPERIMENTALLY, by flipping, that the coin was or was not weighted would take a very tightly contolled experiment with many repititions. Certainly 4 flips is not statistically significant; and flipping by hand is not random.

I personally knew a guy who was prcticed at making coin tosses come out per his wish. He was not perfect, but very good at it. You may inadvertently be doing the same thing. Of course, you can't tell after 4 flips. Try 10,000.

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I have heard that the mathematician/statistician/former professional magician Persi Diaconis is able to do this. –  Greg P May 4 '11 at 1:31
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