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Having an issue with card problem. I feel like I'm getting tricked or something.

Two cards are dealt from a deck, the first one face down and second face up. How many distinguishable outcomes of this experiment are there?

I would guess there is 52 possible outcomes, because the card drawn face-up can be 1 of 52 cards, but the face-down card will always appear to be the same, but something feels wrong about this.

Any help is greatly appreciated.

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    $\begingroup$ It may be unknown to you (until you peek), but it's not the same. It's not the same as the face-up card, and it's not the same as any of the cards not yet dealt.Thus, $52 \times 51$ possible outcomes. $\endgroup$ – quasi Aug 28 '17 at 3:50
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Your problem is English, not mathematics. As I read the problem you are supposed to consider different face down cards as different. The point of the face up/face down distinction is to make it clear that you draw two cards but you care which card is face up and which is face down, so we are asking how many ordered two card sequences there are out of a $52$ card deck. You have answered the question you understood correctly.

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Define the word "distinguishable".

Is it: "Differences exist that are discerned only by observation of revealed cards"?

Is it: "Differences exist that are discerned by examination of the hidden cards"? (And do you need to consider just the face down card, or all of the cards in the deck?)

As to which definition you would need to use to properly evaluate probabilities, well, that depends on what probabilities you are required to evaluate.


There are $52$ (equally) possible outcomes for the face up card.   If the events of concern only involve that card's value, then that's all your model needs to be concerned about.

There are $52\times 51$ equally possible outcomes for both dealt cards.   If the events of concern may involve both cards' values, then that is what your model needs to deal with.

There are $52!$ equally possible ways the deck could have been shuffled.   If the events of concern might involve any card's value, then you would have to consider with this.   Likely not, though.

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