Your wording is fine now, so maybe you already edited it. The key, as JavaFan said, and your post now includes, is that Santa always removes an empty package before giving you the choice of swapping. If he just removed a package randomly, then there would be no point in swapping, because there would be a 1/3 chance that you would be giving up the prize, 1/3 that you would gain it, and 1/3 that Santa would have removed it. Santa is basically allowing you to trade your one package for both of his, while guaranteeing that the prize will be in play. You'd always trade one package for two. It doesn't matter whether he removes an empty one, or trades them both for your one and lets you open both, discarding at least one empty one.
I've had to get out a deck of cards, pulling out a Queen and two Aces, to demonstrate this to people who insisted the odds had to change because the circumstances changed. "Look, let's go through all the possibilities. You're trying to pick the Queen. I deal out Q A A. If you pick the first card, you lose if you switch. But if you pick the second or third card, I discard the other Ace, and you win by switching, so you win 2/3 of the time. Now let's try it with A Q A....." It really doesn't take long, because there are only nine different combinations. Probabilities just aren't taught anymore.
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It's not a matter of just using the wrong words. There's an unstated assumption.
The point is, that to know switching gives a benefit, the candidate must know that Santa will always discard an unpicked, empty, price, and offer you a chance to switch. If Santa barges in, offer you a pick of three, and after your pick he says "well, what if I tell you that this unpicked price is empty, will you switch?", then it may very well be that Santa only offers you a choice if you have initially picked the top price.
Perhaps one should describe the game as:
Santa offers you three parcels. One parcel is filled with chocolate and beer, the others have a goat inside. But from the outside, they all look equal. You get to pick a parcel, but before opening it, regardless which one you picked, Santa will open one of the unpicked parcels, and it will always be a goat. Santa will then always offer you to switch. | [reply] |
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