Conditional probability involving deck of cards.

I had a discussion with a friend the other day:

Deck of cards (52 cards). 13 of them are hearts. Well shuffled, randomly dealt.

__Situation 1__

I'm sitting at a table **by myself** with 2 cards in my hands. Both are hearts.

The rest of the deck is on the table.

X = The probability that the top card of the remaining deck, when turned, is a heart.

__Situation 2__

I'm sitting at a table **together with 4 other people** we all have 2 cards in our hands. I do not know which cards are held by the other players. Both my cards are hearts.

The rest of the deck is on the table.

Y = The probability that the top card of the remaining deck, when turned, is a heart.

__Question__

X = Y ?

And why / why not?

Re: Probability problem - please help! :)

Hello, drwmn!

Quote:

I had a discussion with a friend the other day.

Deck of cards (52 cards). 13 of them are hearts. Well shuffled, randomly dealt.

__Situation 1__

I'm sitting at a table **by myself** with 2 cards in my hands; both are hearts.

The rest of the deck is on the table.

X = The probability that the top card of the remaining deck, when turned, is a heart.

__Situation 2__

I'm sitting at a table together **with 4 other people**; we all have 2 cards in our hands.

I do not know which cards are held by the other players.

Both my cards are hearts. .The rest of the deck is on the table.

Y = The probability that the top card of the remaining deck, when turned, is a heart.

__Question__

X = Y ?

And why / why not?

The answer is YES; the probabilities are equal.

__Situtation 1__

You have two Hearts.

The rest of the deck has 50 cards: 11 Hearts and 39 Others.

. . $\displaystyle P(\text{Top card }\heartsuit) \:=\:\frac{11}{50}$

__Situtation 2__

You have two Hearts.

The rest of the deck has 50 cards: 11 Hearts and 39 Others.

Randomly select six cards from the deck

. . and move them to the bottom of the deck.

(These are the cards dealt to the other three people.)

No matter what six cards were moved,

. . the probability that the top card is a Heart remains the same.