How do I calculate probability of drawing three particular cards out of the deck?

Question

I hope someone familiar with mathematics and probability can help me with this. I have a deck of 78 cards. I draw three cards out of the deck at random. I put them back into the deck and shuffle it. Then I pull out the same three cards out of the deck. What are the odds of this happening? It would be nice if someone could write up how to do this step by step. Thanks!

Answer 1

since you re-shuffle After taking out 3 cards

edited.....

grav below is right

Answer 2

The first time you draw them has nothing to do with the second time if they are randomized inbetween. So we really only need to look at the second.

When you draw one card, the deck has one card less.

So for the first card, your chances are 3/78 for one of the three cards, but your second chance is 2/77. Then the third is 1/76.

Chances:

(3/78)·(2/77)·(1/76)

= 6/456,456

= 1/76,076

Answer:

One in 76,076

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Realistically though, when you shuffle it is rather likely that some set of three cards does not get randomized. If you stick the cards back in the deck randomly or draw cards randomly (or at least differently than you put them back), your chances will approach the figure above more. If you put them back in all together and then draw three cards in a row, you'd have to shuffle extremely well to get the random chance above.

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Response to latest update:

The first draw is really just a "selection". It doesn't matter whether you draw a card or select a card, because your method of choosing a card doesn't affect your chances the second time around.

Your chances of drawing any given card after you shuffle it (assuming the shuffle is random) is 1/78. That figure doesn't change if you choose that card by drawing it previously, guessing a card, having someone else name a card, or any other method, as none of these things affect what card you would draw.

(Realistically, drawing a card and shuffling could affect it since you can't perfectly shuffle it, but that could either guarantee that you always draw the same card or completely prevent it, depending on how you shuffle.)

If you shuffled that card back in a second time and drew another card, your chances of getting that card would again be 1/78. The same would happen the third time and so on.

If you wanted the likelihood of it happening in a row (after choosing what card to expect), that would work like:

(1/78)·(1/78) = 1/6084

Answer 3

just as likely as any other card coming out which is

since the probability of particular card first time is 1/78 at random and since it's random the 2nd and the 3rd time since the cards are shuffled it's

(1/78)*(1/78)*(1/78)