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Poll ResultsPoll 364: Coins, part 2: You toss a coin five times, and it comes up heads 5 times in a row. You don't know if the coin is fair or biased. What is the chance the next toss will be tails?
Total votes: 3785
So, compare to the results of poll #362, which asked exactly the same question, except without the italicised bit:
The point of this pair of polls was to see how different I could make the results of two polls that are essentially identical in what information they give you. I seem to have done fairly well. The addition of the italicised sentence, "You don't know if the coin is fair or biased," doesn't actually tell you anything new that the original question didn't tell you. Its presence does, however, change the perception of how many people see the question, as evidenced by the quite large difference in the results.
In the original question, where the possibility of a biased coin wasn't even mentioned, most of you (at least 82.9%) made the assumption that I was referring to a fair coin - in fact an ideal fair coin, one that has exactly a 50% chance on any given toss of landing on heads or tails. Given the phrasing of the question, that is indeed a reasonable assumption. Usually when one asks probability questions involving coins, there is an implicit (or explicit) assumption of fairness.
However, looking at it from a less ideal context, you might notice that I never specified explicitly that the coin was fair. That was of course deliberate. If it's not specified at all, you can either make the assumption that it's fair (as most of you did), or dig deeper and ask the question, "What if it isn't necessarily a fair coin?"
If you ask this question, you have to look at the information given in the question to try to determine how likely it is that the coin is indeed not fair. A reasonable assumption to begin with is that the coin might either be (a) fair, (b) biased towards tails, or (c) biased towards heads. Given that you know the outcome of five successive tosses, and they are all heads, it should be reasonably obvious (I hope) that it is more likely that the coin is biased towards heads than that it is biased towards tails. You can't figure out exactly how much more likely it is without making some further assumptions*, but the fact that it is any more likely at all is enough for us to work with.
[* You can use Bayesian probability analysis to come up with exact answers, if you make some assumption about the probability distributions of fair and biased coins or the distribution of results of coins. In fact several people did such an analysis and reported it on the forums or to me by e-mail. Their results differed, because their assumptions differed. But they all agreed that the next toss is more likely to be heads than tails.]
These chances of the coin being biased (rather than fair) may be quite small, but no matter how small they are, the fact that the coin is more likely to be biased towards heads than tails means that the next toss is necessarily more likely to be heads rather than tails. Again, you can't work out exactly how much more likely, but any amount at all is enough to make it so that the probability of the next toss showing heads is greater than 50%.
Up to 12% of you correctly followed this chain of argument the first time I asked the question. The second time I asked the question, I changed absolutely nothing about the events described, but I pointed out that you did not know something - something which you in fact also didn't know the first time I asked the question. In one sense, it was exactly the same question. In another sense, of course, I was subtly changing it by pointing out something that you really already knew, or were ignoring by making a different assumption.
With the new version of the question, around 42% of you recognised the implications of not knowing if the coin was fair or biased and made the argument above to select your answer. That's a 30% difference in answers, between what are - arguably - two instances of exactly the same question.
This is not to belittle the 48% of you who still thought the answer should be exactly 50%. Even when it's pointed out that you don't know whether a coin is fair or not, the best working assumption is that it most likely is. Five heads in a row is not terribly unusual for a fair coin, so it's not strong evidence that the coin is biased. The point of the chain of argument above is a fairly subtle one - that if the coin is biased, it's more likely to be biased towards heads than towards tails.
The point of all this is not to examine coin tossing probabilities, but how the way we think about things differs depending on what we think we know about a situation. In both these questions, you didn't know if the coin might be biased or not. The first time around, most of you assumed it was a fair coin, despite that not being stated, because that's what you've been led to expect by your prior experience. The second time, all I did was point out some lack of information. That made many of you question your prior assumptions and think through the question more deeply.
Think about that next time you are asked to make an estimate of what the result of some course of action (by yourself, a relative, a friend, or some political leader) might be.