Odds and Evens

What should you call when you flip two discs to start the game? Evens.

And I can prove it*. (Of course, if you really hate maths, you’ll just have to take my word for it. But it’s not complicated.)

Let’s call the two possible landing positions for a disc heads and tails.

First, let’s consider the case where the disc is entirely ‘fair’ – so that it’s exactly 50/50 which way it lands. Then the four results – HH, HT, TH, TT – are all equally likely, and there’s two odd and two even. So it makes no difference whether you call odds or evens.

Now there’s only one other possibility – that the disc is not fair; that either heads or tails happens more often, on average. That’s equivalent to saying that the probabilities h and t are not equal; or, more usefully:

h-t ≠ 0   [h and t are not the same, so one minus the other won’t equal zero]

Now we can play with the maths a bit:

(h-t)² > 0   [The square of any non-zero number – positive or negative – is larger than zero]

(h-t)(h-t) > 0   [Same thing written out in full]

hh – ht – th + tt > 0   [Just multiplying out the brackets]

hh + tt > ht + th   [Adding ht + th to both sides, which won’t change which side is bigger]

And look what’s happened – the two ‘evens’ possibilities added together are more probable than the two ‘odds’ possibilities. You’ve got to call evens.

Remember we didn’t specify whether heads or tails was the more likely** – just that they weren’t equal – but it doesn’t matter. If the discs have a bias either way, you need to call evens. And even if they don’t, if it’s straight 50/50, calling evens is just as good as calling odds. Evens is never*** the wrong call to make.

*To be fair, this proof is due to Colin Johnstone, an astrophysicist who discovered it while trying to prove the opposite. My wishy-washy intuitive grasp of calling evens didn’t persuade him, but in trying to point out my error he accidentally proved my point. Doubtless he’s still miffed.

** Paul Illian’s spectacularly in-depth article on this topic suggests that which way a disc is biased can depend on the surface it lands on. But it doesn’t matter, when flipping two discs, as long as both discs are biased the same way – evens is always best. Paul also has a nice graph (Figure 8) showing visually that odds is never better than evens. In fact, Paul’s article is just flat-out comprehensive, but I really like the concise 5-line proof given here so I posted it anyway.

*** If the discs are in fact biased in opposite directions (one is more likely tails, one more likely heads) then a similar argument would show that you should call odds. But no matter how scuffed the discs, I don’t think there are many situations where you could reasonably expect two discs with the same shape to behave in opposite ways. Perhaps if you’re throwing on two different surfaces, Paul’s results would suggest that might be the case – but the chances are very high that you’ll be on the same field. Another remote possibility I can imagine is where one flipper is facing upwind, and the other downwind, so that the discs are flipping in opposite directions relative to the breeze. It’s possible that the wind getting under the hollow side of the disc could affect the likelihood of heads/tails differently for the two discs – that would take a lot of experimenting to find out! Instead, just make sure you’re facing the same way and always call evens… 😉