Since there were math questions the other day, there was one that I found very confusing as a kid. The original was in Swedish, and it was quite some time ago that I heard it. So to compensate for inflation, and to make the translation simpler, I’ve used an exchange rate of 1 SEK = $1. (Don’t I wish!) It went something like this:
Three men were hiking in the countryside. It was getting late and they had nowhere to sleep, so they stopped at a farm and asked if they could spend the night in the barn. The farmer said fine, but it’s going to cost you $10. Each. So they paid him $30 and went to sleep.
After they left the next morning the farmer felt bad about charging them so much, so he gave the farmhand $5 and told him to run after the men and give it to them. But while the farmhand was running, he said to himself, “no one’s going to know if I kept some money for myself”. So he kept $2, and when he caught up with the men and gave them $1 each.
So each of the three men paid $9. $9 times 3 is $27. The farmhand kept $2 for himself. $27 + $2 is $29. Where did the last dollar go?
Actually, I still find it confusing. If I take notes, I can see where the story goes wrong. But it sounds so convincing in my head…
I can see it if I draw it, but just like a good magic trick it still gets me when I hear it.
I wonder where it came from originally. As I remember it, my dad (who is a retired math/physics teacher) told it to me, and I may have also seen it in a book of brain teasers. It’s interesting to me that it’s internationally known.
I saw that one as a youngster, too, there are lots of versions of it. Bud cut straight to the point, for the correct solution, adding up where the money actually is. The puzzle is constructed to be as confusing as possible, mixing up numbers that shouldn’t be put together.
By picking the amounts this way, to come up $1 off, it’s more difficult to see where the math goes wrong. If you change the amounts, for example, if the farmer refunds all $30, and the farmhand keeps $3, then using the same math, each man has paid $1, the farmhand has $3, so ($1 × 3) + $3 = $6, and now $24 is missing. Obviously, something is now very wrong with the setup, and the reader would have an easier time figuring it out.
I had a friend who believed the Monty Hall problem. And if others don’t know what it is, my friend claimed that if you have a choice for one of tree doors and then change your mind and pick a different one, you odds increase. I would like to know what the incorrect solution is. Maybe I’m not dumb enough. I think a lot of people don’t realize that what has gone before is of no consequence to what happens next. For example if yyou are flipping coins and 100 heads come up in a row, the chances that the next flip will be tails is 50/50. Unless it’s Sid’s Quarter.
That doesn’t work in the casino (except for roulette). When we sailed to Hawaii last February, the very first day I went down to the casino. One thing I think is true is that they want you to win at the beginning, and they get their money back later on. I went down there and found a row of 5 machines that announced their latest big payouts. There was one that hadn’t yet payed out. I played that one and won about $500. I don’t gamble, because I understand odds. But 45 years ago I was in Monte Carlo at the Grand Casino. The night before I had a dream that predicted exactly how I would win. I played and got down to just a few bucks and then won a jackpot. It wasn’t big, but it let me continue. I got down to just a few dollars and I won a second jack pot. It was bigger. Maybe a hundred franks. I went through that and just before I ran out of money I hit a big jackpot. Flashing lights and whistles went off. The concierge came over and with great fanfare handed me about 10,000 franks, around $1200 at the time. Back then it was a good piece of change.
In a three-door version of the Monty Hall problem, my common sense tells me that it shouldn’t matter if I change door or not, because there is a 50% chance of the car being behind either of the two remaining doors. And my common sense is never wrong, because it also tells me that ice is heavier than water, and that heavy objects will fall faster in a vacuum!
But then someone asked me to imagine a 100-door version, where the host reveals 98 of the doors… Hmm…
So I just wrote a program to simulate ten million runs of the three-door version. If I stuck with the first pick, I got the car 33.3% of the time. If I switched, I got the car 66.7% of the time. So I guess it checks out. I obviously need to get a bumper sticker saying, “I ignored my common sense and all I got was this fancy car.”
I don’t gamble either, but many years ago I used to get quite a few spam emails (written to appear as if they had been sent to me accidentally) where someone was hawking what I discovered was known as “martingale” to win at roulette: Bet $1 on one color, double your bet “plus a little more” every time you lose, go back to $1 every time you win. And he just happened to mention an online casino where I should try it.
Of course it was a scam, but at least I had fun reading about why it doesn’t actually work. (In one of my simulated runs, before I remembered to add a halting condition, I was down by a million dollars but won it all back by betting more than two million. Yeah, that seems likely…)
For whatever psychological reasons that people can’t handle probability well, it’s very tough to get understanding of the Monty Hall problem, and also to explain it once you’ve got it. I read it first in Marilyn vos Savant’s column (in Parade magazine), and it took me a long time to come around on it, through all the arguments that got printed. The 100 door argument was made, but that’s headgames, not math. It was computer modeled, getting the same result that Torbjorn did, children played it with candy and paper cups, and also got the same result. But, as the economists say, “It may work in practice, but how is it in theory?”
My main problem was that the explanation of the theory was never convincing (when I finally took real prob and stat classes, I could do the math with all the conditionals, but that came later). The explanation that I finally came to, which I never saw clearly stated through all the arguments, is: If you pick 1 of 3 doors, you have a 1/3 chance of winning – no matter what options or choices are given after that, if you don’t change, your odds of winning don’t change. It’s false to think that your odds are now 1/2 because a door has been opened, because nothing has changed for your original choice. They could make you leave the room, open the door, land an alien spacecraft, and bring out the dancing girls, but if you stick with your door, it’s still 1/3.
The opening a second door, and offering the chance to change is, in practical terms, saying “Pick a door. Now, do you want to stick with that door, or pick both of the remaining doors?”
I think the real problem is the mindgame aspect, as Monty Hall himself said, they wouldn’t play the game that way on the show. The contestant is sitting there thinking that this must be some kind of trick, which it would have been on the show. But, the problem as given is just pure math, no psychology.
Regarding roulette, in the last stats class I took, Stochastic Processes, we did the Gambler’s Ruin problem, in which the question is to determine an optimal betting strategy to increase $X to $Y, given a certain p(winning) on each spin. If the best odds are less than 1/2, the gambler’s best move is to bet everything, rather than to try to bet small and build a pile. That’s why the green numbers are on the wheel – they knock the odds to less than 1/2, so in the long run, the player always loses.
The strategy of doubling the bet every time you lose is very old, and people keep “rediscovering” it as a sure thing. Again, this would work if the odds were 1/2. And if you had infinite money, and there was no table limit. Also regarding roulette, while casinos take great pains to keep the wheels fair, and prove that they’re fair, they also try to convince the players that they can work out a system. Last time I was in a casino, each roulette table had a display of the last 8-10 winning numbers. I’m told that some casinos used to record and provide printouts for the numbers each wheel had each day. Keeping the hope alive in the players that they can beat the wheel is what keeps the money coming in.
I honestly do not remember this series. Is that a good thing?
It’s from 1994.
Since there were math questions the other day, there was one that I found very confusing as a kid. The original was in Swedish, and it was quite some time ago that I heard it. So to compensate for inflation, and to make the translation simpler, I’ve used an exchange rate of 1 SEK = $1. (Don’t I wish!) It went something like this:
Three men were hiking in the countryside. It was getting late and they had nowhere to sleep, so they stopped at a farm and asked if they could spend the night in the barn. The farmer said fine, but it’s going to cost you $10. Each. So they paid him $30 and went to sleep.
After they left the next morning the farmer felt bad about charging them so much, so he gave the farmhand $5 and told him to run after the men and give it to them. But while the farmhand was running, he said to himself, “no one’s going to know if I kept some money for myself”. So he kept $2, and when he caught up with the men and gave them $1 each.
So each of the three men paid $9. $9 times 3 is $27. The farmhand kept $2 for himself. $27 + $2 is $29. Where did the last dollar go?
Actually, I still find it confusing. If I take notes, I can see where the story goes wrong. But it sounds so convincing in my head…
I remember this one. It’s confusing. But the farmer has 25, the farmhand has 2 and the men have 3.
I can see it if I draw it, but just like a good magic trick it still gets me when I hear it.
I wonder where it came from originally. As I remember it, my dad (who is a retired math/physics teacher) told it to me, and I may have also seen it in a book of brain teasers. It’s interesting to me that it’s internationally known.
I saw that one as a youngster, too, there are lots of versions of it. Bud cut straight to the point, for the correct solution, adding up where the money actually is. The puzzle is constructed to be as confusing as possible, mixing up numbers that shouldn’t be put together.
By picking the amounts this way, to come up $1 off, it’s more difficult to see where the math goes wrong. If you change the amounts, for example, if the farmer refunds all $30, and the farmhand keeps $3, then using the same math, each man has paid $1, the farmhand has $3, so ($1 × 3) + $3 = $6, and now $24 is missing. Obviously, something is now very wrong with the setup, and the reader would have an easier time figuring it out.
I never thought of it that way. Reminds me of the explanation I’ve heard for the Monty Hall problem. Thanks, I’ll have to try and remember it.
I had a friend who believed the Monty Hall problem. And if others don’t know what it is, my friend claimed that if you have a choice for one of tree doors and then change your mind and pick a different one, you odds increase. I would like to know what the incorrect solution is. Maybe I’m not dumb enough. I think a lot of people don’t realize that what has gone before is of no consequence to what happens next. For example if yyou are flipping coins and 100 heads come up in a row, the chances that the next flip will be tails is 50/50. Unless it’s Sid’s Quarter.
That doesn’t work in the casino (except for roulette). When we sailed to Hawaii last February, the very first day I went down to the casino. One thing I think is true is that they want you to win at the beginning, and they get their money back later on. I went down there and found a row of 5 machines that announced their latest big payouts. There was one that hadn’t yet payed out. I played that one and won about $500. I don’t gamble, because I understand odds. But 45 years ago I was in Monte Carlo at the Grand Casino. The night before I had a dream that predicted exactly how I would win. I played and got down to just a few bucks and then won a jackpot. It wasn’t big, but it let me continue. I got down to just a few dollars and I won a second jack pot. It was bigger. Maybe a hundred franks. I went through that and just before I ran out of money I hit a big jackpot. Flashing lights and whistles went off. The concierge came over and with great fanfare handed me about 10,000 franks, around $1200 at the time. Back then it was a good piece of change.
In a three-door version of the Monty Hall problem, my common sense tells me that it shouldn’t matter if I change door or not, because there is a 50% chance of the car being behind either of the two remaining doors. And my common sense is never wrong, because it also tells me that ice is heavier than water, and that heavy objects will fall faster in a vacuum!
But then someone asked me to imagine a 100-door version, where the host reveals 98 of the doors… Hmm…
So I just wrote a program to simulate ten million runs of the three-door version. If I stuck with the first pick, I got the car 33.3% of the time. If I switched, I got the car 66.7% of the time. So I guess it checks out. I obviously need to get a bumper sticker saying, “I ignored my common sense and all I got was this fancy car.”
Anyway, what’s so bad about getting a goat? See https://xkcd.com/1282/
I don’t gamble either, but many years ago I used to get quite a few spam emails (written to appear as if they had been sent to me accidentally) where someone was hawking what I discovered was known as “martingale” to win at roulette: Bet $1 on one color, double your bet “plus a little more” every time you lose, go back to $1 every time you win. And he just happened to mention an online casino where I should try it.
Of course it was a scam, but at least I had fun reading about why it doesn’t actually work. (In one of my simulated runs, before I remembered to add a halting condition, I was down by a million dollars but won it all back by betting more than two million. Yeah, that seems likely…)
I once destroyed workplace productivity for a couple days by introducing the Monty Hall puzzle, as well as the prison lightswitch and logical pirate puzzles (https://www.cartalk.com/radio/puzzler/prisoners-and-light-switch
https://puzzlewocky.com/brain-teasers/the-pirate-puzzle/)
For whatever psychological reasons that people can’t handle probability well, it’s very tough to get understanding of the Monty Hall problem, and also to explain it once you’ve got it. I read it first in Marilyn vos Savant’s column (in Parade magazine), and it took me a long time to come around on it, through all the arguments that got printed. The 100 door argument was made, but that’s headgames, not math. It was computer modeled, getting the same result that Torbjorn did, children played it with candy and paper cups, and also got the same result. But, as the economists say, “It may work in practice, but how is it in theory?”
My main problem was that the explanation of the theory was never convincing (when I finally took real prob and stat classes, I could do the math with all the conditionals, but that came later). The explanation that I finally came to, which I never saw clearly stated through all the arguments, is: If you pick 1 of 3 doors, you have a 1/3 chance of winning – no matter what options or choices are given after that, if you don’t change, your odds of winning don’t change. It’s false to think that your odds are now 1/2 because a door has been opened, because nothing has changed for your original choice. They could make you leave the room, open the door, land an alien spacecraft, and bring out the dancing girls, but if you stick with your door, it’s still 1/3.
The opening a second door, and offering the chance to change is, in practical terms, saying “Pick a door. Now, do you want to stick with that door, or pick both of the remaining doors?”
I think the real problem is the mindgame aspect, as Monty Hall himself said, they wouldn’t play the game that way on the show. The contestant is sitting there thinking that this must be some kind of trick, which it would have been on the show. But, the problem as given is just pure math, no psychology.
Regarding roulette, in the last stats class I took, Stochastic Processes, we did the Gambler’s Ruin problem, in which the question is to determine an optimal betting strategy to increase $X to $Y, given a certain p(winning) on each spin. If the best odds are less than 1/2, the gambler’s best move is to bet everything, rather than to try to bet small and build a pile. That’s why the green numbers are on the wheel – they knock the odds to less than 1/2, so in the long run, the player always loses.
The strategy of doubling the bet every time you lose is very old, and people keep “rediscovering” it as a sure thing. Again, this would work if the odds were 1/2. And if you had infinite money, and there was no table limit. Also regarding roulette, while casinos take great pains to keep the wheels fair, and prove that they’re fair, they also try to convince the players that they can work out a system. Last time I was in a casino, each roulette table had a display of the last 8-10 winning numbers. I’m told that some casinos used to record and provide printouts for the numbers each wheel had each day. Keeping the hope alive in the players that they can beat the wheel is what keeps the money coming in.