The coin box is weird, conceptually.
You have three boxes, one with two gold coins, one with two silver coins, one with a gold coin and a silver coin.
If you take a random coin out of a random box, and it's gold, what chance is there that the other coin in the box is silver?
Works similar to Monty Hall (yes, I can't spell) only with randomness. Conceptually similar, yet the con man in this case doesn't have to influence the outcome of the event to change the odds of the color of the other coin in the box.
Curse you, you quoted my post before I fixed the typos!
That's an interesting one I haven't come across before. I may not sleep tonight.
It took me quite some time to get my head round the two-goats-and-a-car thing, as I said, back in the 1990s when it was apparently discussed in a newspaper. I was told the problem by someone who had read it in the newspaper, and simply informed that some people thought the motivation of the host was important, but my informant couldn't see how. I didn't have internet access at the time.
At first I couldn't see why opening a door could possibly change the 1/3rd chance. I swore blind that there was no advantage to switching. However, I then imagined the 100-doors scenario, and realised in fact that there must be an advantage. Assuming the host is deliberately avoiding the car at step 2, the odds for both doors are now combined onto the one remaining door.
So then I declared that of course one should switch! You'll double your chances!
Then my informant said, but you don't know whether or not the host has deliberately revealed a goat or not. And how come his
motivation can influence the odds?
I got all confused, but realised that it does, again by reference to the 100-doors version. And come to think of it, there are other games I've heard of where a steady profit can be made from a situation where the other player thinks an outcome is 50/50, but the person controlling the choices can skew the odds in his favour by always making the same decision. You don't win every time, but over repeated games, you always come out on top.
Then I thought, the problem has no solution, if we don't know the host's motivation! No advantage to switching if he's not avoiding the car, advantage if he is!
Then, almost finally, I realised that of course that in itself is the answer. Since in neither scenario are your chances
reduced by switching, and there is a greater-than-zero chance that he is playing the deliberate avoidance of the car version, then that in itself tips the odds in favour of switching in your favour.
You can't decrease your chances and you may increase them, so go for it!
That took me about a week to work out, almost entirely on my own.
Then the possibility that Monty is cheating was raised in this thread....
There's always a wrinkle you haven't considered.
Hence my final formulation of the answer,
if you can be assured that Monty is not cheating, then switch.
So who knows how long it will take me to figure out the coin boxes.
Rolfe.
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