The beginning of many Ultimate (nee, Frisbee) games is marked by flipping discs to decide which team must pull (kick off) and which goal each team will defend at the start of the game. This is sort of like the coin flip before an American Football game. Two players -- one from each team -- flip a disc in the air. A third player -- a representative from one of the teams -- calls "same" or "different", referring to whether both discs land with the same side (top/bottom or heads/tails) facing up or different sides facing up. If he guesses right, his team gets to choose whether they want to pull or receive to start the game or if they would like to choose which end of the field to defend.
From my casual observations, most players tend to pick "same" or "different" quite randomly. That would be all well and good if the probability of each event were equal, but they are probably not. If the probability the disc lands top side up (we'll call that heads) is equal to p and the probability it lands bottom side up (tails) is equal to q, then we can calculate the probability of the same and different outcomes of two flips of the disc. The probability both discs land heads up is p2 (assuming the two flips are independent), and the probability that both discs land tails up is q2. That means the probability of "same" is p2+q2 because the two outcomes are mutually exclusive. The two discs can land with different sides up if the first disc lands heads up and the second lands tails up (this occurs with probability pq) or if the first disc lands tails up and the second lands heads up (with probability qp). Because these two events are also mutually exclusive, the probability of "different" is 2pq.
Given those two equations, we can calculate the probability of "same" and different" for different probabilities of heads and tails. I have graphed the probability of same and different versus the probability of heads, as shown below.
As you can see, the probability of "different" is maximized when the probability of heads (and tails) is equal to 0.5. That maximal probability, however, is only 0.5. Therefore, if the probability of heads and tails are equal, you have a 50% chance of winning the flip if you chose "different". But if the probability of heads and tails are not equal, the probability of "different" drops below 50%. That means there is never a scenario in which choosing "different" is a smart bet (greater than 50% chance of winning), but choosing "same" is a smart bet unless heads and tails are equally likely (and, even then, it's even money).
So, I say to all ultimate players out there, ALWAYS CHOOSE SAME.
UPDATE: See the follow-up post.
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Aren't you assuming that p (and q=1-p) are the same for both discs? But isn't it more reasonable to assume that, while no disc has a perfect p=0.5 probability of landing 'heads', the p's of no two discs are likely to be the same? (Assume, perhaps, that each disc's p is drawn independently from some kind of larger distribution, maybe dependent on something like manufacturing or the way the person throwing it holds it, before throwing).
Are you sure your analysis holds in this more general case?
The key assumption here is that the frisbees and the throws are exactly the same. If one is more likely to land up and the other more likely to land down, then "different" would be a better bet. It seems reasonable to conclude that the actual result is somewhere very close to 50-50, or people would have caught on quite some time ago and the system probably would have been abandoned as nonrandom. In other words, empirical data suggest that the model is incomplete. A great example of how an elegant mathematical model may fail to reflect reality because it oversimplifies real life conditions. :-)
I have a follow-up post where I show that choosing SAME is a good bet if the deviations from 50/50 are in the same direction for both discs. If the deviations are in opposite directions, then different is the safe bet.
All discs used in ultimate, both officially approved and otherwise, tend to have the same shape. The shape is determined by two factors, aerial stability, and not destroying your hand when you try to catch it. The shape could be described as a very flattened dome or the leading edge of an airplane wing swept into a circle. The bias is likely in the same directon for anything shpaed like this.
Besides, virtually everyone in the ultimate community uses the Discraft Ultrrastar (and the approved Frisbee brand disc is virtually identical), so any differences a redue to wear. Also, from experience, any differences that might change the polarity of the bias, would render the disc usable only as a dog bowl.
Now, if someone showed up with a golf disc, this all goes out the window. (Which, by the way, is the tee shot for my favorite par 3)
I think that having the bias the same is a sufficent assumption, but I believe that the least-restrictive assumption that is necessary would be that the bias is in the same DIRECTION for each disc, although I'm to lazy to try working through it for a general case.