Monday, May 28, 2012

Chances Are, We Don't Understand Chances

Personally, I think probability is one of the most fun math concepts to teach. Break out the dice, the coins, the different-colored marbles, and the spinners. Do a bunch of trials to see how the experimental compares to the theoretical.

Despite the fun, I see a lot of students get all the way to high school without a solid understanding of what probabilities really mean. Take, for example, these two questions:

#1 You flip a fair coin five times and get five heads in a row. What's the probability of getting heads on the sixth flip?

#2 What's the probability of flipping a coin six times and getting heads all six times?

People often think these are asking the same thing. Our gut instinct for #1 is that if we've already gotten an uncommon five heads in a row, surely the chance of getting heads again isn't that good. But the coin doesn't know what it landed on before. The situation only has two choices: heads or tails. For that single sixth flip, it has a 50% chance of landing heads just like every other time.

The situation in #2 is completely different. You're taking all six flips as one situation, so there are a lot more "choices" for the results. All heads, all tails, one tail and five heads (with six different configurations for this one alone), and so on. There is only a 1/64, or a little more than 1.5% chance, of this happening.

The difference in the two is that in #1, the five heads in a row have already happened, and cannot influence the sixth flip.

It's also good to talk about what makes a game fair or unfair, and why gambling isn't such a great idea.

The thing about probabilities is that they often make an assumption about all else being equal. The coin or dice being evenly weighted. Every individual outcome (like heads or tails) having an equal chance.

In life, we can't always make that assumption. That's where people sometimes confuse "probability" with "statistics." For example, say we collect some data and find that 2% of writers querying a novel this year will secure representation with an agent. Does that mean any given querying writer this year has a 2% chance of getting an agent?

Not remotely.

Within that pool of querying writers, we can't say "all things being equal," because they aren't. Some of the writers don't have a clue what they're doing. (You've seen Slushpile Hell, right?) Some aren't making such egregious mistakes, but just aren't ready yet. Some just don't have the right timing with market trends. Some aren't querying that aggressively, only sending out a few here and there. And then some are at the top of their game, do their homework, and go at it. The percentage of that last group getting representation is probably quite different.

So, strange as it is for a math teacher to say, don't get caught up in the numbers when it comes to these subjective, highly variable, real life scenarios. Save thoughts of probability for when you're deciding whether to walk into a casino, or figuring out whether you should take an umbrella when you leave for work.

When it comes to situations where all things aren't equal, work to make sure you belong to the group that successes draw from. That's the way to up your chances.


Anonymously Yours said...

After I got past the twitching as I remembered all of those math classes I passed by the skin of my teeth, I found you have a great point.

I will give myself kudos...I knew that the first two questions were different. Couldn't tell you why, just knew they were! I have a math prof that would be proud...

E.B. Black said...

This post really made me happy to tell you the truth. Thank you.

T.L. Bodine said...

As much trouble as I've always had with math, probability was always one of my favorite topics. I'm also absurdly delighted by statistics.

I forget where I got my data, but I once determined that any single unpublished book's chances of becoming an NYT best seller are roughly equivalent to the odds of the Apophis meteor striking the earth.

Lucky for us, best sellers happen every day, but planet-altering meteors seem a bit more rare :D

Mindy McGinnis said...

Ah yes... it helps if you know what you're doing :)