how to game a fair coin

From Jean-Paul Sartre:

"Everything has been figured out, except how to live."

Statistics 101

Here's a common setup: someone flips a coin once. What are the chances that it lands heads?

The reflexive answer is to say "50%", and that of course would be academically correct.

The more skeptical response would be a question: "Is the coin fair or is it rigged?"

Here's the second setup: someone flips a coin 50 times, and it landed heads every single time. What are the chances that it lands heads on the 51st toss?

The reflexive answer might be "pretty high, maybe 75-80%?" Which would, academically, be wrong.

As we know, a fair coin's probability of landing heads or tails is not dependent on what it flipped before. That is, tossing a coin is an independent event. Whether you tossed it 50 times before and got heads every single time doesn't affect the chances of it landing heads or tails that 51st time-- the odds of it landing "heads" is still 50%.

But if you view the flipping of a coin 51 times as a sequential series of events, you get a weird situation indeed. Probabilistically, the chances of a coin flipping heads once is 50%. But the chances of flipping heads two times in a row is equivalent to the product of the probability of heads on each toss, which would be 50% x 50%, or 25%. 

And flipping heads 51 times in a row? The chances are incredibly low (0.5^51, or 0.000000000000004%). On its own, that 51st flip has a 50% chance of landing heads or tails, but the chances of 51 flips in a row being the same face are damn near zero.

So a statistically-minded person would say "The flip itself: 50%. But 51 consecutive heads: 0.000000000000004%." And a skeptic would say "Let me see that damn coin, it's got to be rigged." 

Both of these responses are fair. But all of these responses rely on a certain view of the assumptions involved in the simple flip of a coin. These assumptions can be broadly recognized as:

  1. Initial Conditions
  2. Launch Conditions
  3. Flight Conditions

The strictly academic "flipping a coin" exercise focuses on only a small subsection of these assumptions. Namely: is the coin perfectly weighted? Note that we haven't examined some of the more interesting assumptions, some more outlandish than others. Do atmospheric pressures fluctuate at all in between tosses (gravity, drag forces, etc)? Does the coin land on a surface that favors a particular head? Is the person flipping the coin strangely skilled at controlling his toss? Does the coin have free will? Can the coin change its own trajectory as its being launched and as it is landing? But in our pedantic conversation about this coin, we've only identified consecutivity and coin fairness as the primary movers of probability.

The Hot Hand Fallacy

The same academic focus is used to describe the "Hot Hand Fallacy." This phenomenon is used to explain the case of a basketball player "catching fire" and shooting lights out, seemingly unable to miss. 

When a player has a hot hand, the reflex might be to say that the probability of his next shot going in is much higher than average. He's on fire, and our intuition suggests that there's something special going. But as we saw in the case of flipping 51 coins consecutively heads, the probability of a prolonged hot streak is substantially lower with each additional shot. At the same time, though, if each shot is an independent event, the probability of making the next shot remains the same as it was for every other shot, despite the hot streak. Mind fuck!

There are many more variables at play here that complicate this even further. (Stanford professor Jeffrey Zwiebel discusses some of them in a recent paper.)

First is in "initial conditions" -- has the player been getting more sleep, or meditating; has the team changed its offense in some significant way, or has it settled into its groove mid-season? 

Or in "launch conditions" -- has the player been working with a shot guru and thus reached a breakthrough in the rotation and range of his shot; is the player, now much more selective with his shot selection, more patient with when and from where he shoots; now that he has a hot hand, are defenders double-teaming him or otherwise changing how they guard him? 

Or in "flight conditions" -- is the player in the middle of a West Coast road trip where maybe the weather in the month of March favors his shot; does the height of a hoop in some West Coast arenas vary by a couple favorable centimeters? Absurd, yes, but still worth considering. 

One man's lucky streak might be another's actual, legitimate hot hand. Until defenses grow accustomed and adjust. Or in Steph Curry's case, don't adjust (because they can't figure out how to). A coin flip isn't just a coin flip.

Loaded Coins

The point of all this is that maybe there is a way to game a fair coin. There are probabilities that surround us that we take simply at face-value that might be more under our control than we may imagine. Elon Musk once stated this as the "branching of probability streams":

"You're going to generate some error between the series of steps you think will occur versus what actually does occur and you want to try to minimize the error. That's a way that I think about it. And I also think about it in terms of probability streams. There's a certain set of probabilities associated with certain outcomes and you want to make sure that you're always the house. So things won't always occur the way you think they'll occur, but if you calculate it out correctly over a series of decisions you will come out significantly ahead."

In some domains, the initial conditions largely determine the outcome, while in others, the launch or flight conditions have outsized impact. But when we assess the probability of certain events or outcomes, we often make assumptions on the nature of the variables involved in these calculations. Yes, many values are fixed, but not as many as we would think. I think we can rig many of the coin tosses in our life.

(I'm not a statistician so if any of this is wrong, please let me know.)