What Are The Odds?

The other night I was watching Stephen Colbert being interviewed on Larry King and he was talking about how a bunch of members of his family were killed in a plane crash when he was young, and Larry King asked him if that gave him a fear of flying, and Stephen Colbert replied saying that it made him less afraid of flying, because after all, what are the chances that airline tragedy would strike the same family twice?

I was thinking about that again today, and then I remembered my 8th grade math teacher, Mr. Hesser. Mr. Hesser was a big guy, he was the wrestling coach, and when he saw his beloved wrestlers walking down the hall, he would throw chalky erasers at them. This was kind of a big deal, because one wrestler in particular, Jimmy Gerovac, was pretty much always wearing black heavy metal shirts, and so a blast of chalkboard eraser dust represented a significant threat to his headbanger ethos.

Mr. Hesser had slicked hair and was always kind of sweaty. I remember thinking that he was a pretty good teacher, because he made math real, but he made math real by making fun of people who couldn't do math. Mr. Hesser (although he seemed to be kind of a country boy himself) in particular loved to make fun of dumb country folk and their inability to do math. He would talk about about how dumb country folk would actually hire him to come out and calculate how much grain would fit in their silos because they had no idea how to do it themselves. "I mean, anyone with an 8th grade education should know how to calculate the volume of a cylinder plus half a sphere of the same diameter, right?" Today, given about half and hour and the proper formulas, I think I could figure it out. I hope.

So what does this have to do with Stephen Colbert?

See, this one time, Mr. Hesser was making fun of dumb country people and he was talking about how most folks wouldn't buy lottery tickets at a gas station where someone has already won a bunch of money recently, because "ya know, what are the chances of someone winning in the same place?" he would say in an exaggerated drawl (on top of what I considered to be his already prominent Hoosier accent). He said they were stupid and superstitious because the chances of the next big lottery ticket being there are still as good as they would be anywhere else. When you roll a six on a die, what are the chances that you will roll a six again? One out of six. Same as the first time. The die doesn't remember where it was last. The chances of someone rolling a one and then a six are the same as someone rolling a six and then a six (1 in 36, right?). The chances of someone rolling three sixes in a row? The same as someone rolling, oh, say, a two, then a four and then a five. By this logic, Stephen Colbert is just as likely to die in a plane crash as anyone else, as each flight is a new roll of the die, unaware of previous statistics.

But. But. Here's the thing. According to this (pretty interesting), the average American's odds of dying in a plane crash in a lifetime are 1 in 5,051. So, if you think of it as a 5,051-sided die, then the odds of rolling the same number twice are 1 in 25,512,601. Pretty unlikely, right? But you can look at it two ways: The odds of one person dying in a plane crash are 1 in 5,051. The odds of a second related person dying also at a later point are still the same, but the odds that both things must happen are really low. But it's still just as likely to happen to a related person than it is to anyone else. The more I think about it, the more confused I become. Wait. This is kind of blowing my mind. Anyone care to step in? Am I totally wrong here?

I mean, his reaction to the question may have been a joke, I don't even know if his reaction was from the point of view of Stephen Colbert the person or Stephen Colbert the character (in which case, he might have been taking the anti-Hesser everyman's approach to probabilities for effect). If it's real, if thinking that way helps him fly the friendly skies, then that is ok. Regardless of the fact that it is statistically faulty logic, I think that there is something to be said for humans believing that a rare thing that has happened once will be even less likely to happen again. After something really bad has happened, I can recall saying to myself, "well, at least I can be pretty sure that things won't be that bad ever again" but it's actually not true! Luckily, the same rules apply for good things, so that's cool too, I guess.

OK mathletes. Prove to me that you aren't as dumb as the bumpkins Mr. Hesser made fun of:
There's a corn silo. It's 20 feet in diameter and perfectly cylindrical up until 35 feet, where it is topped by a half-sphere of the same diameter. How much corn (in square ft) can you fit in this darn thing? Show your work. First person to answer gets a prize. Super bonus points if you can tell me how much all that (dry) corn would weigh.