Problem 938: Exhausting a Colour

[ConstantClover]

A deck of cards contains R red cards and B black cards.
A card is chosen uniformly randomly from the deck and removed. A second card is then chosen uniformly randomly from the cards remaining and removed.

• If both cards are red, they are discarded.
• If both cards are black, they are both put back in the deck.
• If they are different colours, the red card is put back in the deck and the black card is discarded.

Play ends when all the remaining cards in the deck are the same colour and let P(R,B) be the probability that this colour is black.

You are given P(2,2) = 0.4666666667, P(10,9) = 0.4118903397 and P(34,25) = 0.3665688069.

Find P(24690,12345). Give your answer with 10 digits after the decimal point.

I am a beginner, and this is a problem with a difficulty rating of 5%. I cannot get started because I cannot get an answer that matches the first clue, P(2,2) = 0.4666666667.

One computation that I tried produced an answer of 5 / 12 or 0.4166666667. Could there be a mistake in the problem statement?

Or maybe I am confused about the order of drawing cards, putting cards back in the deck, and determining when the game is over. Here are a couple of scenarios that I hope someone can clarify?

1) Deck: 2R, 2B; Draw: 2B -- After drawing two black cards, there are only two red cards remaining in the deck. Is this game over, or do the two black cards get shuffled back into the deck? This would lead to an infinite series, but I cannot rule that out.

2) Deck: 1R, 2B --
a) Draw 1: R -- Could it be that the game is over after drawing only one card?
b) Draw 2: B -- Like 1) above, after drawing two cards, there is only one in the deck, so this might mean that the game is over.
c) Shuffling R back in... Deck: RB, play continues...

Are cards that are drawn still considered "in the deck"? Are we expected to only determine the end of the game before the draw? I would like to assume the answers to be affirmative, but then I do not know how to handle the infinite series.

Or maybe the number given in the clue is wrong?

[SAG145]

The game ends when all cards are the same color, but after both cards have been drawn and returned/discarded.
The word "removed" does not indicate that they are not part of the deck after they have been drawn, but that the same card cannot be drawn twice.

Regarding the infinite series, the game could be infinite with probability 0, so that's not really a possibility.
I think this is a relatively hard 5% problem, and I would suggest you start with easier problems.

[ConstantClover]

Thanks, SAG145!

And there is no error in the question, I take it?

[hacatu]

It's not wrong, it's always possible that there are multiple reasonable interpretations but I feel it's pretty well worded

You seem to be thinking about drawing the cards 1 at a time, but that's not really how the game works. "Draw 1 card uniformly at random and then 1 uniformly at random from the rest" means "pick a pair of cards, equally likely among all pairs" or however else you want to phrase it.

The wording maybe is a bit confusing, because when you "remove" these cards, you are really just looking at the pair you draw. Then you either return the cards to the deck (so you didn't really remove them, just looked at them), discard them (so now they really are removed from the pool of remaining cards), or return one and discard the other.

[ConstantClover]

Thanks, hacatu!

I appreciate your comments, and I understand that finding the correct interpretation is sometimes the crux of the puzzle. My problem here is that no interpretation I try gets me the correct answer.

Unfortunately, the rules of this forum seem too strict for me to describe what I have tried. I do not see a path forward. Is there something more that can be said without violating the rules?

[ConstantClover]

Good news!

I am now getting a number that makes sense -- as indicated in the question P(2,2) = 0.4666666667.

Whew! Thanks for the help, everyone!

*case closed*