The oppressive nature of mathematical facts

I was frustrated watching the news tonight, seeing reports of long lines of people waiting to buy lottery tickets for tonight’s $500,000,000+ jackpot. There are several angles of absurdity to consider here. For example, why all the excitement over $500 million? Wouldn’t any of these players be equally thrilled by $10 million or even $5 million? Another example: Several of the people interviewed, who had just bought hundreds of dollars of tickets, were unemployed, facing foreclosure, struggling to pay medical bills, and so forth. But putting all that aside, let’s just look at the math.

Sometimes, when presented with large numbers, we simply fail to grasp how large they are. This is especially true when it comes to matters of probability and statistics. So let’s try to break this down into smaller pieces to make it more tangible. What if I asked you to guess my home address? I would even tell you what street I lived on and you knew there are only about 100 homes on my street. (Pretend you had a list of valid address, so you won’t be able to guess a home address that doesn’t exist.) It’s possible you could guess it on the first try, but we all know it would be relatively unlikely. The odds would be 1 in 100.

We can grasp that. 1 in 100 is a tangible probability. Those are pretty good odds, right? Well, it depends on the context. Put those odds it another situation: what if you went to the doctor and were diagnosed with an illness that had a 1 in 100 chance of survival. Would you like those odds? No, in that context 1 in 100 is very bad. It is a very slim ray of hope, right? Kind of depressing.

Oh, but it gets worse. What if I only told you what city I lived in? Now the odds are 1 in 4734 (all statistics are from the 2010 census). Now what if I only told you that I lived in Ozaukee County? Odds now shift to a meager 1 in 36,000.

Let’s keep going … Now you only know I live in Wisconsin: 1 in 2,634,781. We all agree this is practically impossible, right? Think about it: all the houses on each road, in each city, in each county, in the entire state of Wisconsin. There is no way you could reasonably believe that you had any real chance of guessing my address given these odds. Yes, there is a chance you could guess it. That’s how probability works: the event can happen. But when you put it in a tangible context and you can begin to visualize the odds of guessing, it looks incredibly bleak.

Here’s the thing: your odds of guessing my home address, armed with only the knowledge that I live in Wisconsin, are still MUCH better than your odds of winning the Powerball lottery. Those odds, according to the official Powerball site, are 1 in 170 million. Let me write out those zeroes: 1 in 170,000,000. Did your eyes just glaze over? Let’s make it tangible again: You have better odds of guessing my home address if all you know is that I live in the United States. There are over 132,000,000 housing units in the US. Do you now see how ridiculously improbable it is that you will win the lottery?

I hear some rebuttals: But I’m going to buy more than one ticket. Surely that will boost my odds.

Yes, mathematically speaking, your odds will get better. But practically speaking, you still have no chance. Let’s put some of the earlier odds in the perspective of buying multiple lottery tickets:

So if you want to make your chances of winning about the same chances of surviving a terminal diagnosis with 1 in 100 odds, just buy 1,750,000 tickets. Does that put in perspective? Oh, and Powerball tickets are $2 each, so that’s a $3.5 million investment to have the kind of odds that would send you home sobbing from the doctor. Depressed yet? You should be. That is the oppressive nature of mathematical facts.