What is the probability that a half dollar coin lands on heads? 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.
 A: 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.
A: 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.
A: 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.
