Thing is, knowing that’s what people would usually do means that any smart person would do the opposite. It seems odd that most people would think the scenario part of the way through, and not all the way.
What’s also odd is the version of the test not shown: where one person repeats the test over and over, with there always being an offer to switch. Even then, it takes humans longer to figure it out than it does birds. Even when we know it’s random, we still stick to the hypothesis that they’re trying to trick us.
There’s no reason why it wouldn’t be. It’s just not the point of that particular experiment. They were only testing the “do people stay or switch” part of the myth. I suspect they just wanted a more visual demonstration of “why it’s better to switch”, which they provided nicely with the comparison between Adam and Jamie.
Yeah, the visual part of these experiments is part of what makes Mythbusters so successful. There have been plenty of urban legend shows in the past, none of which lasted very long because they just told you whether something was true or not. It’s plenty easy to just say “It’s better to switch,” “The airplane will take off,” or “The toilet will/will not blow up.” But having the visual of all those red squares really sells the myth.
I thought they did a very good job with it, actually. I’ve read the Straight Dope column a couple of times and probably a few of our threads on the problem, but the answer never really sank in. The machine was a good demonstration - and of course when they finally got around to explaining the math it was extremely simple.
What he said was that they couldn’t compare the success rate of the staying strategy against the switching strategy because nobody chose the switching strategy so they had no data. So they ran the second experiment where Jamie used the staying strategy while Adam used the switching strategy.
But their assumption was incorrect. Because the two strategies mirror each other, you can make the comparison even if you only have the results from one strategy. The success rate of one strategy will always be the complement of the other.
Exactly. That’s what I thought. Just look at how often they won by staying, and subtract that from 20 to see how often they would have won by switching. Simple.
For anyone else having trouble, here’s a data set. In order to get started, here’s what you have to do:
Create a list of 15 “games.” For each game, choose the letter A, B, or C. That letter represents the door the treasure is behind.
Compare your list of 15 games to my list of guesses in the spoiler boxes below.
For each game, eliminate a letter that is neither on my list nor on your list. That letter represents the door that you open and show me does not contain a prize. (For example, if, for game 7, you chose B for the prize, and I guessed A, you’d have to eliminate C. If for game 8, you chose B for the prize, and I guessed B, you could eliminate A or C.)
On your first run-through, understand that I’ll switch my choice after your elimination. (For example, if I chose A, and you eliminated C, I’ll switch my choice to B).
Give me a point for each correct answer.
If you don’t agree that this models the Monty Hall problem, please explain why. Step 3 is the most difficult step to understand: keep in mind that Monty will always open a door (thereby eliminating it as a possible choice), and he’ll never open the door with a prize or the door the contestant initially chose.
If you do agree, then set up your list of 15 games, and then compare them to the spoiler boxed guesses below as described in the rules above.
[spoiler]
C
B
A
C
A
B
B
A
C
A
C
B
C
B
A[/SPOILER]
If you want a control set of data, run everything again, only this time I won’t switch. For yet more sets of data,
Use the set above, but start on a different turn number
When I explain this to (high school) students, who are often stuck on the “it must be a 50/50 chance” idea, I usually put it this way: imagine I’m in a boxing match with Mike Tyson. There are only two possible outcomes, he wins or I win. Those outcomes certainly are not equally likely.
Of course, some of these little whippersnappers don’t know who Tyson is, so I sometimes try a different tactic. Almost all my students play some sort of sport. So, you’re on a soccer team. Either your team wins or the opposing team wins. If there’s a 50/50 chance, there would be no reason to practice, study plays, or work out. The whole point of those is that they affect the probability of winning.
I also agree that it looked like Adam had more wins than I would expect, but the sample size was fairly small. If they had repeated things a few thousand more times, the win rates would be pretty close to the theoretical values.
It’s always seemed pretty simple to me. There’s a 1/3 chance the prize is behind the door you picked. So there’s a 2/3 chance it’s behind one of the other doors. The prize certainly isn’t behind the door Monty opens, so there’s a 2/3 chance the prize is behind the remaining door. So you switch.
I saw Alan Davies and Marcus du Sautoy stage it using toy cars and plastic farm animals. Alan lost.
A subtle but very important point to remember is that all of this only applies if the game show host is always, no matter what door you pick, going to open and reveal another, empty, door. As long as he’s always doing that automatically, then it’s just a question of math.
then the problem becomes one of psychology and so forth, and if you’re someone who’s purely math-oriented, then a sufficiently clever game show host might realize that, and might do the monty hall thing ONLY if you’ve in fact chosen right initially, knowing that you are thus going to assume that the puzzle math is relevant, and that he will thus fool you into switching.
Yes. Change the rules, and you have a different problem. Again, that’s why it’s so important that the conditions are stated specifically. The classical Monty Hall problem has the host always open an empty door. That’s when it becomes an interesting problem, an apparent “paradox”, and a lesson in how probability works, instead of… well, some random stuff involving doors.
In the episode, Adam won 38/49 (a 77.5% win rate) whereas Jamie won 11/49 or a 22.4% win rate.
The expectation is for Adam to win 66.6% and Jamie to win 33.3%, but the sample size is small and I suspect their rig may not have been properly randomized.
During the episode, they said they randomized the doors on the stage, but if they just “randomly” stuck the stickers on the long paper sheet, that might not have been as good as a true random number generator.
As far as the psychology of it, a contestant walking into a three-door choice set would not know that the host would open a door, or that the opened door is known to be empty. If the revealed door is random, then switching to win really is 50/50. You also could be shown right away that you lost.
Sure. So what? The host *could *also suddenly start farting out golden turkeys, or the door could be a magic portal leading to Narnia. But that’s not the Monty Hall problem.