You are one of a hundred people in a contest to win a new car. There are one hundred keys in a large bag but only one of them will start the car, winning it for the contestant who draws that lucky key. Each contestant will get only one draw and the order will be determined by lottery...except for you. You can choose what position you will draw in, but you must decide before the drawing begins. What position do you choose? Think about it just briefly before reading on. At least decide whether you'd prefer going first, last or one of the middle positions.
The first one to draw is the only contestant guaranteed to have a chance at the winning key so that seems like the best choice at first. Reasoning from the gut, it seems improbable that the first draw would win. Going last certainly seems foolish; there's an almost certain chance the car will be taken by then. So, maybe waiting a little while for the first ten or maybe twenty people to fail--as is likely--will somewhat improve your odds. Sure, it's not a great trick or anything. It might improve your odds from 1 in 100 to 1 in 80, but that's better than nothing. Some math will tell you the exact right answer, but hold off on that. Just take an approximate guess at which position maximizes your odds.
If you're like most people I've asked--including myself--you bought my casual arguments and guessed a position slightly earlier than when you would be surprised that it wasn't gone, like 20 to 30. By 50, you'd expect it to have gone already, so you want to win before that happens.
I call this the Reverse Monty Hall Problem, so let me briefly digress to explain the infamous Monty Hall Problem. There are three doors with a prize hidden behind each. A bag of gold lies behind one while the other two contain a goat. You choose a door and then one of the other doors is opened, revealing a goat. You are then asked if you want to switch your choice of door. Most people feel that they have gained no new information (we already knew one of the two unchosen doors had a goat) and so feel no need to switch. The choice appears a meaningless option. I personally argued vociferously with my statistics professor in defense of that opinion. Even though switching literally doubles the odds of success, it felt completely intuitive that it was pointless.
I call the car lottery question the Reverse Monty Hall Problem because it does the opposite of the original: it makes people think they have meaningful control when they do not. Your position in the draw order is completely meaningless; each position has a 1 in 100 chance to win. An equivalent situation would be one where the keys were given out randomly and you were allowed to choose when to try your key. It wouldn't matter. Either your key works or it doesn't.
The math couldn't be much simpler, yet in the small group of people I've tried this on, the mistaken belief that there was some element of control ran pretty strong and it took a while to clear it up. Particularly, the argument that "only the first person has a guaranteed shot at winning" seemed to resonate with everyone and implied there was further, more complex, math to be done to analyze it. Everyone had to conquer a gut feeling that this seemingly powerful option had no effect.
I imagine this behavior is well known to casino managers. While gamblers have much meaningful control over what they do, almost all persistently believe in the concept of being "due to win" or games " going cold" even though it would likely be illegal to implement that way. I imagine a large majority of people trying to flip "heads" on a coin would choose one that had just (fairly) flipped 10 "tails" in a row over one that seemed more random.
I wonder how often these factually incorrect biases affect us. From the powerful effects I have seen elicited by these problems, I venture there are many instances in our lives where we fail to capitalize on very large opportunities. While the Monty Hall Problem is masterfully contrived, it is very simple and its effects are not at all subtle (doubling your odds!). What real world choices do we make where we discount equally valuable information? We must exclude instances where we simply don't yet have the right data; we can always do more research before making a decision. And the chooser must agree that the unchosen outcome is superior.
I also wonder, on the other hand, where we waste energy on endeavors that do not, in our own opinion, significantly enhance our lives. It seems easy to come up with numerous examples, but usually some thought will tease out their hidden value. For instance, basketball players are not required to dribble before shooting a free throw but almost all do. I doubt that pointing out that they were expending unneeded energy would alter their behavior. Dribbling before free throws has value to them, likely the same benefit of any ritualistic behavior. I'm looking for a case where someone exerts unwanted effort for literally no return and they would immediately stop doing it if they knew. Or, from another perspective, where they invest effort in influencing a system they believe is truly random. Note that this excludes gamblers who believe they can influence the cosmos with their will but would includes gamblers who play purely by the numbers.
