Tuesday, January 18, 2005

"I don't believe in probability"

I think it's my purpose in life to kill off crappy ideas before they poison too many weak and vulnerable minds. Sort of a flu shot for bad reasoning. If I discover crap being spread, I put a stop to it, or I at least don't let it float by uncontested. Kind of why I like The Daily Show.

Some defenses of spreading terrible reasoning that I'd leave alone, out of charity and good will:

"I didn't think about it, I just passed it along"
"I thought the terrible reasoning was obviously flawed; I spread it to be funny in an ironic way"
"I am too dim to figure these things out on my own"

Some defenses of spreading terrible reasoning that I can't accept:

"I don't believe in probability"

The reason I can't accept this defense is because it's defending crappy reasoning with more crappy reasoning, which, being naughty in my sight, I must snuff.

Let's consider an example.

"Since there is no such thing as an average coin flip, we can't say anything about flipping coins than involves using the idea of an average coin flip."

The premise is certainly true. Since the expected value of a coin flip is 50% heads, 50% tails, there's no such thing (at least on a standard coin) as a result that is average. I'll even grant that I've never seen an average coin flip. And while there are some people of average age, and average annual income (assuming we round to some level of accuracy), there aren't folks of average gender. Well there may be, I suppose, but that's complicated. So we'll stick to coins.

So the premise is true (no such thing as an average coin flip), but does the conclusion follow? Is it true that since there are no average coin flips, all reasoning based on the idea of an average coin flip is flawed?

Let's find out.

Let's say I offer a bet. To play, one must put up $1000. I flip a fair coin 100 times. If the number of "Heads" results is between 40 and 60, inclusive, I pay the player $2000. If not, I keep the $1000. And one can play as many times as one wants, and we settle up when one is done playing.

Should one play?

If we decide to deny the existence of probabilistic reasoning, then we really can't answer that question with anything more than a gut emotional response.

Of course, if we're rational folks with a grasp of probability and math, or at least know someone who is, we can figure this out. But it requires us to accept one little thing:

Using the notion of an average coin flip can lead to valid conclusions.

If we accept this idea, then we discover that the expected number of "Heads" is equal to

(Number of flips) * (Average number of heads per flip) = 50

To figure out the odds of getting a number between 40 and 60, we can either use the cumulative binomial probability distribution function (BINOMDIST in Excel), or we can use the idea of a standard deviation.

Standard deviation of the binomial random variable = (Yikes! More "average coin flips"! ) =
((Number of flips) * (Average number of heads per flip)* (Average number of tails per flip))^1/2 = (100*.5*.5)^1/2 = 5

So when we want to know what the odds are of getting a result within 2 standard deviations (50-40, 60-50 = 10, 10/5 = 2), we can use a standard Z table to figure out that there's a 95.4% chance of getting such a result.

So the average game (no such thing as average games?) pays $2000 95% of the time, and pays $0 5% of the, so the game, on average, pays one $1900 to play. And it costs one only $1000 to play. Should one play?

Well, if one doesn't believe in any of this "average coin flip" and "average game" and "average American" nonsense (as one shouldn't, cuz, heck, has anyone ever seen an average coin flip? And one either wins or loses, one doesn't 95% win), then one really can't know what to do. But if one understands that the idea of an average coin flip, auto accident, voter, stock market outcome, American, etc. Is a valid concept in probabilistic reasoning, then one can make some money on the game.

And this perfectly sound bit of thinking, along with our confidence in the readily available data from the CDC, is what lets us say things like:

"The average American is 11.6 times more likely to die from a gun than from a medical mistake."

We can be done now, or the free thinking lessons can continue.