Flipping Coins and Cancer CasesA random column.

July 18, 2013  4:00 pm

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 appliedmath 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 memoryless. Random things with this memoryless 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 5050. 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 1/2^{n}. 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.
Cancer Epidemiology Biomarkers and Prevention" journal's Fred Hutchinson Cancer Research Centre's report =
2,000 women (5574 age) on statins for over decade got >2 times more invasive ductal lobular breast cancer (type is ~10  15% of breast carcinomas).
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.
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."
Plus, Math Net:the Case of the Swami Scam.
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 56 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 coinflipping 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.
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.
Once is happenstance, twice is coincidence, treat a third instance as enemy action.
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 5050, 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.
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.
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?
http://www.radiolab.org/2009/jun/15/