Odd Lesson - Odds and the LHC

Today's lesson is a lesson in odds. And in how oddly people respond to odds.

First of all, what do we mean by odds? Odds have two meanings. One meaning is a ratio of probabilities. Suppose you are going to make a bet with a friend on the roll of a fair six-sided die. If the die rolls a 1 or a 2, you win. If it rolls a 6, your friend wins. The probability that you will win is twice the probability that your friend will win, so the odds are 2 to 1 (or 2:1) in your favor.

Gambling odds, on the other hand, tell you the ratio of how much you would win (if you were to win) to how much you would lose (if you were to lose). If you and your friend are playing fair, the gambling odds will be equal to the probability odds, in this case, 2:1. That is, if you win $1 for a dice-roll of a 1 or 2, you should have to pay your friend $2 for a dice-roll of 6.

In this example of a fair bet, the probability of you winning times the amount you win (1/3*$1) is equal to the probability of you losing times the amount you lose (1/6*$2). If you and your friend were betting on a hundred-sided die, and your friend wins on a roll of 1, and you win otherwise, then you'd better give her 99:1 odds. If you stand to win $1, you should stand to lose $99.

In each case above, the amount that you expect to lose, weighted by the probability of losing, exactly balances the amount that you expect to win, weighted by the probability of winning. Another way of looking at it: If you played the game many, many times, you are likely to win as much as you lose, and come out even.

(Note: Bookmakers and casinos do NOT give you gambling odds that are equal to the probability odds. If you play their game many, many times, you are likely to lose more than you win. That's how they make money.)

From the description above, it might sound like anything that pays you better than a fair bet would be a gamble most people would be willing to take. This is certainly true for small bets, but when the amount you stand to lose is not $2, but a substantial fraction of your worldly goods, you might hesitate to risk it even if the odds are way in your favor. For example, a wager in which I win $1 or lose $100,000 is a fair bet if the probability of me winning is 100,000 times as large as the probability of me losing. However, I would not be willing to take the bet, even if the odds were even more in my favor, say 200,000:1. Someone, like me, who fears losing large amounts, even at low odds, is said to to be risk-averse.

Insurance companies take advantage of our risk aversion. Suppose you pay $500 a year for insurance on a $100,000 house. You could make a gamble, and not pay for said insurance. Your winnings are then $500 (you come out $500 ahead by not paying for insurance), but the amount you stand to lose would be $100,000. That would be a fair bet, if the odds of losing your house in a year were 200:1. They aren't - the odds are much more in your favor. However, most people aren't willing to risk losing their house, even if the probability is very small.

Risk and odds get even more interesting when you are talking about non-monetary things. Your life for example. How much would you pay to avoid risking your life? Take safety belt use as an example. There is a small, but not negligible, probability that using a safety belt could save your life. Are you willing to make the following gamble: risk losing a few seconds buckling your safety belt in order to win your extended life? What probability odds would make that gamble worth the risk? In this case, too, I am quite risk-averse, and I wear my safety belt.

You may have heard people say, "I've driven a hundred times without my seatbelt and never got killed." Of course, we all know that the 101st time could be the one that gets them, but they are trying to make a statement about their measure of the odds. From this small amount of data, it looks to them like the probability is vanishingly small. However, drawing statistical conclusions from a small data sample like that is not a good idea.

Now let's considering something even more precious. More precious than your life? Yeah, like the destruction of the world. That's a huge thing to risk. Suppose you had to make the wager where if you lose, you lose the whole world, but if you win, you win a greater understanding of elementary particle physics. What probability odds would you need to be willing to take that chance? 2:10⁹ per year?

The problem with this question is that the probabilities are so unimaginably small, and the consequences are so unimaginably large that it is difficult to get your brain around them.

You may recognize the gamble I have just described. A lawsuit against the Large Hadron Collider (LHC) tried to shut it down on the basis of the possibility that black holes or strangelets produced could destroy the earth.

Many a scientist has blogged about how ridiculous/insane the guy was for suing. Part of their argument against the suit sounds a lot like the guy who doesn't wear his safety belt. The argument goes like this: We've been hit by a large number of cosmic rays as energetic as the particles in the LHC, and we haven't lost the earth yet.

Don't get me wrong, I'm all for the LHC, and, of course, the statistics we get from cosmic rays are very good because of the large numbers. In order for our seatbeltless driver to get this kind of statistics, he'd have to have driven 100,000 times a day every day for the history of the earth. Yeah. In spite of all this, the risk-averse coward in me has to show some understanding for those who fear large losses even at unimaginably small probability.

P.S. Scott Adams has an amusing take on the risks involved in the LHC. See this post on the Dilbert blog.