In doing a little research for my last post on innumeracy I stumbled across a site on mathematical fallacies. An excellent site, I highly recommend giving it a once over.

But on the site I found the following definition and example:

The Law of Averages thinking- A belief by gamblers that the more often you win or lose the more likely your luck will change on the next try. If you flip a coin and it lands heads 10 times in a row, what are the odds that it will land heads on the 11th try? Answer 1:1. What about after 100 times in a row? Again it is 1:1. The odds are the same on each toss!

I agree completely with the definition, I do not agree with the example.

The key to the fallacy is “your luck will change”. If you believe the fallacy and you flip a coin and it lands heads 10 times in a row, then you’ll think that the next flip is more likely a tails. The fallacy is that because you know the average is half heads and half tails, and you have had far more heads than tails, then a tails is due soon. I’ve heard this a lot, but it’s definitely a fallacy. Assuming the coin is fair and the trials are independent, the probability is still 50%. Just because you’ve seen a bunch of heads doesn’t mean you’re due for a tails.

But here we’re back to a strange place. These trials aren’t independent, even if the process does not imply a dependence. Statistical independence is not the answer to the question “Are event A and event B causally related?” but rather “Given that event A has happened, can I change my mind about the likelihood of B?” In our example, we just got 100 heads in a row, that doesn’t change your idea of the likelihood of heads on the next flip?

If I walk up to you with a quarter and flip it 100 times in a row in front of you, and every time is a heads, are you actually willing to bet even money that the next flip is a tail? The probability of getting 100 heads in a row with a fair coin on independent trials is less than 1 out of 10^30. That’s a 1 with 30 zeros after it. You have a better chance of winning the lottery 4 times in a row. So you’re going to tell me, you’ve just witnessed an event similar to predicting which photon on the sun will hit my left index finger’s cuticle in 8 minutes, and you don’t think something is just a little bit rigged? You still believe the trials are independent and the coin is fair? Then man do I have a bet for you.

Most people have heard about the study that discovered that a quarter isn’t fair. There is a slightly more favorable chance of getting a heads than a tails. How’d they discover this? They flipped a coin lots of times, and it came up heads more than tails. In our example, it came up heads every time. You’re crazy if you don’t change your opinion of the probability of the likelihood of heads. Empirically, it’s not 50%, it’s 100%.

Now, does that mean it’s impossible to get a tail? No, not at all. It means if a head comes up, we’re not surprised at all, and if a tail comes up, it’s tantamount to finding the cure for cancer. But as soon as a tail comes up, you should probably change your idea of the probability that the next flip will be a tail. It’s now a little more possible, given our prior experiments.

Probability is all in what you know. I know that there are two options, lighting strikes me when I walk outside in a thunderstorm, or lightning doesnt strike me when I go outside in a thunderstorm. That means there’s a 50% chance of getting struck by lightning when I walk outside in a thunderstorm. But years of experimentation have led me to believe that’s not true. I don’t go outside all that often in a thunderstorm, but every time I have, I haven’t been struck by lightning. I’m willing to bet that even though it’s much less than 50% likely that I’m gonna get struck by lightning, it’s probably still a dumb idea to go running out in a thunderstorm.

If you believe the fallacy, then I’m due for a lightning strike. If you believe the lightning is “fair” then you’ll still say it’s 50/50. But if you’re willing to adjust your idea of the likelihood of getting struck by lightning because of empirical evidence, then you’ll not only think that the probability is fairly low, but the other two people are complete morons.

All this talk reminds me of late nights with you teaching me probability on the phone.

Posted with : Bare with Me

haha, and you thought I was doing you a favor helping you out with your probability. I just sit around thinking about this stuff all the time anyway.