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 *p*^{2} (assuming the two flips are independent), and the probability that both discs land tails up is *q*^{2}. That means the probability of “same” is *p*^{2}+*q*^{2} 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 2*pq*.

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.