Andrew Gelman alerted us to a classical probability error that showed up on the Freakonomics blog recently. (by way of Xian).
Probabilistic thinking is not very intuitive. One wishes the Freakonomics team would attack conventional wisdom, instead of embracing it. Here, they trumpeted the "very long odds in the Israeli Lottery": the fact that the same six numbers won the same lottery three weeks apart. This feels like an extremely rare event but it is a case where we tend to underestimate severely its rarity.
My readers may want a refresher of what the mistake is.
Let's start with the much simpler "Birthday Problem". Say you're in the jury selection room with a hundred people. What's the chance that you can find two people born on the same day? The chance is much higher than you would imagine! In fact, the math shows that it is more or less 100%! Even if you have 60 people in the room, it is almost certain. With only 30 people in the room, there's about a 65% chance of finding at least two with the same birthdays.
The key to understanding this puzzle is to realize that we did not fix the day of birth; we are not looking for two people born on January 1st. The two people born on the same day could have been born on any one of 365 days (excluding Feb 29 for convenience). The chance that we can find two people born on exactly Jan 1 is much less likely than the chance of finding two people born on the same day.
Similarly, there are 4,950 distinct pairs of people in the room, any one of which can be the pair of people with the same birthday. So, there are many ways in which we can find two with the same birthday and that's why it is much more likely than we think it is.
The point being: If you take enough drives in the safari, you will find a leopard.
Turning to the Israeli lottery. The "rare" event in question can also happen in many ways: the numbers could reappear one, two, three, ..., ten weeks apart (assuming it's a weekly lottery); and the reappearance could happen in any one of I'd think thousands of lotteries all over the world. In fact, every single time a lottery is drawn anywhere in the world, there is a chance that the outcome will prompt some journalist to write up the same story: in X lottery, the same Y numbers won Z weeks apart, what long odds!
The point being: The more drives you take in the safari, the more likely you will eventually find a leopard.
Perhaps the difficult part of taking this argument is: why should we be thinking about these imaginary scenarios if we already know that it happened in the Israeli lottery and it happened three weeks apart? Doesn't the reality make the imaginary scenarios irrelevant?
Here is how I'd think about this: if you already know that the Israeli lottery drew the same numbers six weeks apart, then the question of what are the odds is silly because the odds are 100% since we know for sure that it happened!
The question of "what are the odds?" makes sense only at the moment when we don't know if something would happen or not. That means one of many possible scenarios could happen.
This need to place ourselves in a state of uncertainty is what I think trips up a lot of people. We do not like uncertainty.
Because of the same reason, some people (most mainstream journalists, it seems) buy the government line that we "made money" or "did not lose money" from the TARP rescue of failed too-big-to-fail banks.
This analysis is done from a state of certainty, from hindsight, taking into account the banks thus far survived the danger. But evaluating TARP from this view is like celebrating a lottery win after you found out you are the winner. The fact that you won the lottery does not change the fact that economically, it was silly to play the lottery in the first place.
When a decision is made in a state of uncertainty, it has to be evaluated in that state of uncertainty. For it is the uncertainty that imposes the greatest cost. In the case of TARP, the key question is to determine whether the government negotiated the right terms on behalf of the people at the time of the decision, when the banks faced enormous existential uncertainty. There is little question that the government did not get a good deal. Just take a look at the kinds of deals that the banks themselves get when they save failing companies from bankruptcy! (This often involves the banks owning a vast majority of the shares of said companies in addition to multifarious onerous conditions.) Or, how banks act when we don't pay our credit card or loan bills: is it okay for us to stop paying for a year, repay the amount in full, and declare no harm done?
Of course, there were a zillion other policies not called TARP that directly benefited banks at a cost to the people, and thus a variety of arguments why the bank rescue was not costless. My favorite is the suspension of mark-to-market accounting so banks can list their assets at fictitious prices (some of these assets are cash flows from "liar loans" and so on). I will outsource the big picture to economist Dean Baker who does a great job "beating the press" every day (here, and here).