The Ask Marilyn column in Parade magazine featured this question and answer in 1990 and 1991. It received over 10,000 responses and over 1000 from PhDs. Can you solve it and do you think Marilyn is right? (yes, this is an old topic but I just encountered it for the first time 🙂 )
Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?
Craig F. Whitaker
Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?
Read the ensuing archived discussion and rebuttals here:
What a simple problem, right? So simple, you don’t need any math training to solve it correctly. Chances are, though, your brain won’t get it right and even after you read the explanation it won’t stick and/or you won’t quite feel right about it.
That’s probability for you. There’s a bevy of books and research materials out now on uncertainty, probability and chance. From The Black Swan to The Drunkard’s Walk to Chance to Fooled by Randomness. It seems to be a hot topic whenever the markets appear to be haywire, when finances get tough, when wars carry on, when we start the red vs blue discussion.
Decision theory, quantum physics, analysis of behavior, managing your daily life, parenting… all involve studying probability of events. Frankly, we’re bad it. We’re bad at researching it and we’re bad at really understanding it. We’re really bad at talking about it.
Not only does the above type of problem vex us, we fall prey to the “if I recognize any detail about a fact, I’m more likely to agree with that fact than something I know nothing about” situation. Same Names, same birthdays, cities you’ve lived in, color of your skin, same employer, political party…. we tend to those things we know even when it might not help us.
Some evolutionary psychologists will offer the explanation that we evolved into taking chances on partial information that are familiar with because it aided in survival more often than not. hmmm.
To me, that’s a stretch. I think we just suck at probability. There’s no evolutionary advantage or disadvantage. It simply might just be that way because of the way everything else about us is put together.
OR, the big OR.
It’s really about behavior. We are conditioned by the environment. We gravitate to people, things, situations we have behavior for – we’ve been punished or reinforced for. When we haven’t yet experienced the other options, we can’t attribute any value to them. If we don’t value something, we can’t chase it or pick it. As soon as we pick or don’t pick it and we experience what happens, we then can assign value to it for the next encounter.
So, why do we suck at probability calculations? We don’t behave based on calculations. We literally do not connect the dots via probability. We associate stimulus (sounds, words, smells, information) with other stimulus. When the connection is made, it’s not a probability statement. Why that is biologically, I don’t know. I know that you don’t get partial action potentials and partial connections and partial patterns in the nervous system. We get patterns and if the stimuli/situation matches a pattern (neural, muscular, etc.) we associate it.
In the game show problem above, it’s uber important to the math and to the behavior to understand the actual sequence. You pick an option first then the host picks a losing option. You’re learning at each step and assigning value – people over value the option they pick first (don’t switch!), the other option is unknown.
You can research this on that old clips of Let’s Make a Deal or just watch Deal or No Deal – people over value their own case they chose and they associate all sorts of crazy data in choosing cases – the strategies mostly stink.
So… our ability or inability to accurately determine probabilities may not be some cause or end to itself but is simply an intervening variable or side effect of how we learn. Now, why we learn the way we do is a big discussion and is worthy of writing about behavior every day!
I sure wish I evolved a better way of gambling… I’d be rich!