Biased coin puzzle

Suppose you have a coin where the probability of heads equals 1/pi =~ 31.8309886%. Next, suppose you and a friend can't agree on which movie to see so you use this coin to choose. To be fair, each of you should have a 50% chance of winning. Multiple flips are allowed.

Question 1: Name any method which will result in a probability of success of 50%.

Question 2: Name a method which will equal or beat a method I know of that requires an average of 4.61 flips to have an outcome.

First person to answer question 2 wins a beer.

As usual, please put answers in spoiler tags.

The question for the poll is which statements do you agree with (multiple votes allowed).

Edit: for the record, I think my hands are invisible to those bathroom faucets.

Also, I saw a TV ad for those Delta kitchen faucets where you tap the faucet anywhere, say with your wrist or forearm, and the water turns on. What happens if you're not at home and a fly lands on it?

Dunno about #1... Dunno about #2 but my answer for a beer:

Question one:

Question 3: What is the shape of a coin that has probability of heads equal to 1/pi?

Quote: MathExtremist

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The method you know is Von Neumann's method: flip pairs of coins until there's a difference, then pick the first. For 1/pi the average number of flips is 4.61.
But here's a paper about more efficient methods; rather than typing it out:
http://www.eecs.harvard.edu/~michaelm/coinflipext.pdf

This was the answer I was looking for. I didn't know it was called what you called it. That paper explains a way to slightly decrease the expected flips.

(Beers I owe you)++;

I also totally agree about those touch-less faucets. To me it is like playing a slot machine, hoping water will come out. For me, the hit frequency is about 25%. There must be some trick to it that those in the know aren't sharing.

Okay, somebody else submit a good math or logic puzzle for a change.

We each pick either heads or tails. If we both pick the same, I win. If we both pick differently, you win. No need to flip the coin.

Another method, not sure if lower than 4.61 expected flips -- but we both flip the coin once. If one person gets heads and the other tails, the one who got heads wins. If both get heads or both get tails, then flip again -- continue until one gets tails and the other heads.

Draw parallel lines with a pen separated by a distance equal to its length. Throw the pen, and if it lands on the line, toss the coin. If it comes up heads, it's a winner. This happens 50% of the time. No need for floating-point arithmetic!