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by
Charlie Martin

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July 18, 2013 - 4:00 pm
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Since science is all about making things predictable, it is sort of a surprise that many of the advances in science in the last hundred years have been made using mathematics about things which are inherently and intrinsically unpredictable: the mathematics of probability, and its applied-math stepchild, statistics.

The usual example of something that’s inherently unpredictable is flipping a fair coin. Take a quarter from your purse, flip it, and it comes up either heads or tails.

Now, because I know my readers, I can tell someone is getting set to write me a comment about how there’s no such thing as a perfectly fair coin, or that it can also land on an edge, or explaining how they learned to flip a coin so it made exactly one turn and so they could always predict how it would land, so let me just say: this is mathematics, it’s a perfectly fair coin, and we’re going to be catching it in the air so it never lands on edge. So just stop.

As I said, when you flip this perfectly fair coin, it either comes up heads or tails. The next time you flip it, it also comes up either heads or tails, and which comes up doesn’t depend on the previous flip at all. Technically, we’d say it “has no memory”, it’s memory-less. Random things with this memory-less property are going to be important, so remember the word.

The gambler’s fallacy is imagining that something like a fair coin actually has memory — in other words, if you’ve had a run of heads, you’re “due for” tails to come up. The truth is that every time you flip a coin, what comes up is independent of all the previous flips. What makes you think you’re “due for” a tails is that over many coin flips, the likelihood of getting a run of many heads or tails gets smaller, and it gets smaller quickly.

Let’s start with the simplest case. If you flip a coin exactly once, the chances of getting all heads are exactly 50-50. It’s either heads or tails, which we’re going to represent as 0 for heads and 1 for tails. Flip the coin twice, and the chance of getting all heads drops to 1 in 4: 00, 01, 10, 11. Three times, and it’s 1 in 8: 000, 001, 010, 011, 100, 101, 110, 111. I won’t carry out the examples any further, but it’s easily shown that this pattern carries on forever, and the chance of getting a run of heads of length n is exactly 1/2n. Now think about flipping a fair coin many many times: for every run of 10 coin flips, we won’t get a run of 10 heads 1023 out of 1024 times. So you’re right that you learned to expect that you won’t get ten heads in a row; the fallacy is that if you have gotten nine heads in a row, you’re still going to get that tenth heads exactly half the time.

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All Comments   (24)
All Comments   (24)
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People's intuitions are often wrong, mine and statistics is bad, but I guess I've seen worse. Ahem. IF an event is random, THEN one pattern does not imply another. But if you hand out 10,000 radium bracelets and 1,000 people show up with cancer, guess what, that is not a random situation, and the other 9,000 people might want to take warning.
39 weeks ago
39 weeks ago Link To Comment
Random ?
Cancer Epidemiology Biomarkers and Prevention" journal's Fred Hutchinson Cancer Research Centre's report =
2,000 women (55-74 age) on statins for over decade got >2 times more invasive ductal lobular breast cancer (type is ~10 - 15% of breast carcinomas).
39 weeks ago
39 weeks ago Link To Comment
Um, yeah, and? I'd like to read the study, but there is indeed a small probability that those results are indeed random. With n=2000 it's probably quite small, but it's certainly greater than zero.
39 weeks ago
39 weeks ago Link To Comment
The basic idea of the post is valid -- people's intuition is wrong and so the masses are too eager to cry foul.

But we must address the challenge of proper belief: As I get more heads in a row, I must shift from the idea the coin is fair to the idea that perhaps it IS biased. That's the basis for credibility theory, and the two types of error.

Real life distinction -- Steve Jobs and Lady Gaga pleased millions, deserve their billions. E. Stan O'Neal and Charlie Prince merely seduced a dozen board directors, and do not.
39 weeks ago
39 weeks ago Link To Comment
Yeah, absolutely. I'm a confirmed Bayesian, and so my belief in the fairness of the coin decreases with each successive head; however, that belief never decreases to zero, because there's always a finite probability that is it, ie, 1/2n > 0 for all n.
39 weeks ago
39 weeks ago Link To Comment
dammit, that should read 2 to the power of n. I've just learned another bit of html the comments filter.
39 weeks ago
39 weeks ago Link To Comment
"The reality is that when things happen randomly, just like our coin flips, the outcome will almost always seem unfair."

When I was a kid and complained that something wasn't fair, My Dad would tell me "Life isn't fair, never has been, never will be, get used to it."
39 weeks ago
39 weeks ago Link To Comment
Thanks, Charlie. I've read the first part of your article many, many times. But not the second part where you point out how many of the apparently 'unfair' outcomes are actually the most common. I've observed that but never understood it - only knowing that in reality the expected 'fair' result doesn't happen that often.
39 weeks ago
39 weeks ago Link To Comment
Sophisticated beliefs of natural evil are rare.

Plus, Math Net:the Case of the Swami Scam.
39 weeks ago
39 weeks ago Link To Comment
Charlie Martin -

1. I think it's preposterous to suppose that the average person would find anything other than (5H,5T) to be unfair, but I'll assume for argument that it's true. There are 252 ways to get (5H,5T). You claim that a person will only see 5-6 of them as fair. I think that's preposterous.

2. It is not true that the outcome of a coin flip can not be controlled. Prof. Persi Diaconis of Stanford University has shown a coin-flipping machine that can flip heads (tails) 100% of the time, if the initial orientation of the coin is heads (tails), and he says that magicians have learned to control coin flips. He finds that if the angle (on launch) between the normal to the coin and the angular momentum vector is less than 45 degrees, the coin never turns over. I see no reason to believe that a human could not learn to keep the angle less than 45 degrees.

3. When something apparently extremely unlikely occurs, such as one person winning the lottery twice, there is no guarantee at all that this is "random." The event may be extremely unlikely in the whole population, but this tells us nothing about its likelihood in particular circumstances, such as cheating or anything else.
39 weeks ago
39 weeks ago Link To Comment
1. Okay. It is, however, still true that that's what people expect. For an amusing hour, see the "Stochastisity" episode of RadioLab Paul has linked in another comment.

2. I quote: "Now, because I know my readers, I can tell someone is getting set to write me a comment about how there’s no such thing as a perfectly fair coin, or that it can also land on an edge, or explaining how they learned to flip a coin so it made exactly one turn and so they could always predict how it would land, so let me just say: this is mathematics, it’s a perfectly fair coin, and we’re going to be catching it in the air so it never lands on edge. So just stop."

3. You're right. But when something extremely unlikely occurs, there's no reason because of that to assume it wasn't random.
39 weeks ago
39 weeks ago Link To Comment
Nevertheless.

Once is happenstance, twice is coincidence, treat a third instance as enemy action.
39 weeks ago
39 weeks ago Link To Comment
1. I doubt very much that it's what most people expect.

2. You don't have to quote yourself. I read the article. If you make "this is mathematics" bear all the weight, then you've got a point - you didn't mean to be talking about the real world. Otherwise I don't think so.

Diaconis found that with naive human flippers it doesn't matter whether the coin is "perfectly fair" (perfectly balanced); it can be very badly unbalanced, and still show 50-50, e.g., with a jar lid as a "coin"; it seems that nothing matters except the angle between normal vector and the angular momentum vector. If memory serves, he also found that "caught in the hand" is less random than "let fall on the floor," and this has nothing to do with ending up on edge.

3. It depends on what you mean by "assume." If you mean "take it as pretty much guaranteed," you're right. But this doesn't change the fact that "particular circumstances" are more likely to be responsible for the event than if its frequency in the whole population weres greater.
39 weeks ago
39 weeks ago Link To Comment
1. I don't care.
2. I was trying to be subtle about pointing out you're being obnoxious; the point is that a fair coin *is* a source of a uniform distribution, or alternatively that a uniform distribution models what we think of as a fair coin. And I specifically pointed out that someone was sure to bring up expert coin flipping etc. Don't expect me to be very respectful when you do.
3. Nope. That's the gambler's fallacy for you. See also my discussion of the null hypothesis.
39 weeks ago
39 weeks ago Link To Comment
"If you make "this is mathematics" bear all the weight, then you've got a point - you didn't mean to be talking about the real world."

Of course he's talking about the real world, because most events like coin tosses are as random as makes no practical difference. Your posts here have been an exercise in obtuse pedantry.

Because every one has a 100% accurate coin tossing machine, right?
39 weeks ago
39 weeks ago Link To Comment
I agree. It's not surprising that a carefully crafted machine can flip a coin and catch it in midair so that it lands the same side up every time. But even this assumes it starts the same way up every time. Same for a magician who tosses and catches a coin like a pizza so it doesn't flip over. That's not really flipping a coin at all. As Persi Diaconis himself says: "The caveats and analysis also point to the following conclusion: For tossed coins, the classical assumptions of independence with probability 1/2 are pretty solid."
39 weeks ago
39 weeks ago Link To Comment
So, if the guy to the left of me goes down with a bullet, and the guy to the right of me goes down with a bullet, then I guess I'm pretty safe, huh?
39 weeks ago
39 weeks ago Link To Comment
On average...
38 weeks ago
38 weeks ago Link To Comment
That's an example of what I meant by "particular circumstances" in my messages to Charlie. (You're at a place that's taking gunfire.) OTOH, the one goes down, other goes down, does improve the likelihood that you're safe. For example, maybe the shooter only means to kill those two guys, not you; or maybe there's something about you that makes you harder to aim at; or maybe the shooter only had 2 bullets left, and in any case his supply has been depleted by 2 bullets, making it more likely that he runs out of bullets without managing to hit you.
39 weeks ago
39 weeks ago Link To Comment
The gambler's fallacy is a really powerful thing.
39 weeks ago
39 weeks ago Link To Comment
I think the main reason it's so powerful is that few people have studied probability far enough to know that unlikelihood of (say) 10 heads can be explained purely by combinatorics, i.e., by the fact that there's only 1 way to get 10 heads, but 1023 ways to get something else.
39 weeks ago
39 weeks ago Link To Comment
I'm posting this related link on "Stochasticity" for a friend who's too lazy to register.

http://www.radiolab.org/2009/jun/15/
39 weeks ago
39 weeks ago Link To Comment
If I had a dollar every time I reminded someone that random events cluster I wouldn't need to play the lottery.
39 weeks ago
39 weeks ago Link To Comment
Yeah. Hopefully I gave some idea of *why* they cluster.
39 weeks ago
39 weeks ago Link To Comment
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