I was playing my flashcard game on my phone, and in the “Mathematics and Measurements” section, I got this question (paraphrasing): “If you toss a coin ten times and it comes up heads each time, is it more likely to come up heads the eleventh?”
And I started thinking.
I know the “correct” answer. The answer is No. If the coin is truly random, each toss is independent of the previous ones, and each toss has a 50/50 chance of coming up heads. In any random sequence of sufficient length, pseudopatterns appear, and if you get one of those it may seem like… well, a pattern. But it’s not. In fact, if you flip a random coin an infinite number of times, it’s essentially guaranteed that any sequence you can think of will come up at some point: ten heads in a row, a hundred heads in a row, the lyrics to “Never Gonna Give You Up” spelled out in Morse code (with heads being dashes and tails being dots). It doesn’t matter how many times it came up heads before: each flip is still 50/50.
The question, as worded, is not about a theoretical coin, a perfect 50/50 randomness generator. It’s about a real coin in the physical world. So I started thinking: If a coin comes up heads some number of times in a row, it’s reasonable to be suspicious that the coin is weighted. What is that number?
I posed the question on Facebook — and damn, was that conversation interesting. People first started talking about p values (very roughly: the odds that a pattern is statistically significant). In much of the scientific world, an experimental result is considered significant if the p value is less than 0.05 — i.e., if there’s a better than 5% chance that the result are statistically significant. But to some extent, that’s just a convention, and for any given situation, you have to consider how much uncertainty you’re willing to accept. (If I’m flipping a coin that came up heads X number of times, and ten bucks is at stake, I’ll accept a higher p value than if my home and all my possessions are at stake.)
But then we got into yet another question, which shifts the answer in a completely different way: How common are weighted coins?
Ten heads in a row is wildly unlikely — but weighted coins are rare, and the odds of having one in your pocket are also wildly unlikely. You then have to do a Bayesian analysis of the question, factoring in the odds of both. If weighted coins were common — if, for instance, the Mint started issuing quarters where Washington’s head is enormous and puffy — I wouldn’t need as many heads-in-a-row coin flips to make me suspicious of this one.
And once you start asking “How common are weighted coins?”, you then have to ask, “Who’s flipping the coin?” If I’m flipping the coin myself, and I just take it out of the change I’ve gotten from various vendors throughout the week, all I need to do is the Bayesian analysis above, based on how many weighted coins are in circulation. But what if someone else is flipping the coin? We then move the question out of statistics, and into psychology, sociology, and game theory. Who’s flipping the coin? How trustworthy are they? Do they have anything to gain from winning the coin toss — and if so, how much? If the person doing the tossing is a stage magician, I’m more likely to think the coin is weighted, even thought weighted coins are generally rare. Ditto if they’re, say, Paul Newman in The Sting.
I don’t really have a conclusion to this story, except to say that this question, which at first seemed fairly straightforward, turned out to be a lot more interesting, more complicated, and more personally variable, than I’d imagined. Thanks to my Facebook friends for a great, thought-provoking conversation!
And yes — I have a peculiar notion of what constitutes “frivolous.”
Frivolous Fridays are the Orbit bloggers’ excuse to post about fun things we care about that may not have serious implications for atheism or social justice. Any day is a good day to write about whatever the heck we’re interested in (hey, we put “culture” in our tagline for a reason), but we sometimes have a hard time giving ourselves permission to do that. This is our way of encouraging each other to take a break from serious topics and have some fun. Check out what some of the other Orbiters are doing!