Question

The Jack of Spades, Jack of Hearts, Queen of Spades, and Queen of Hearts are taken from a deck of cards. The four cards are shuffled and two cards are selected from the deck (without replacement).

Let A = "Both of the cards you selected are Queens." For (A) - (D), give ?(?)P(A) under each of these conditions. All these problems are to be considered separately.

(A) Suppose the first card is a Queen.

(B) Suppose that the second card drawn is a Queen, but you don't know what the first card is. (So, the point is, is this different from A?)

(C) Suppose at least one of the cards is a Queen.

(D) Suppose that one of the cards is the Queen of Spades.

(E) Suppose when you draw the first card you throw it away without looking at it. You don't know what it was. What is the probability that the second card is a Queen? (Note that this is NOT the same problem as(B).)

Answer 1

A={(Queen of Spades,Queen of Hearts),(Queen of Hearts,Queen of Spades)}

(A) Let B=event that first card is Queen

B={(Queen of Spades,Jack of Spades),(Queen of Spades,Jack of Hearts)(Queen of Hearts,Jack of Spades),(Queen of Hearts,Jack of Hearts),(Queen of Spades,Queen of Hearts),(Queen of Hearts,Queen of Spades)}

(B) C=event that the second card drawn is a Queen={(Jack of Spades,Queen of Spades),(Jack of Hearts,Queen of Spades),(Jack of Spades,Queen of Hearts),(Jack of Hearts,Queen of Hearts),(Queen of Spades,Queen of Hearts),(Queen of Hearts,Queen of Spades)}. Hence A is a subset of C.

(C) D=event that at least one of the cards is a Queen={(Queen of Spades,Jack of Spades),(Queen of Spades,Jack of Hearts),(Queen of Hearts,Jack of Spades),(Queen of Hearts,Jack of Hearts),(Jack of Spades,Queen of Spades),(Jack of Hearts,Queen of Spades),(Jack of Spades,Queen of Hearts),(Jack of Hearts,Queen of Hearts),(Queen of Spades,Queen of Hearts),(Queen of Hearts,Queen of Spades)}

(D) E=event that one of the cards is the Queen of Spades={(Queen of Spades,Jack of Spades),(Queen of Spades,Jack of Hearts),(Jack of Spades,Queen of Spades),(Jack of Hearts,Queen of Spades),(Queen of Spades,Queen of Hearts),(Queen of Hearts,Queen of Spades)}

(E) F=event that the second card is a Queen={(Jack of Spades,Queen of Spades),(Jack of Hearts,Queen of Spades),(Jack of Spades,Queen of Hearts),(Jack of Hearts,Queen of Hearts),(Queen of Spades,Queen of Hearts),(Queen of Hearts,Queen of Spades)}.

P(F)=6/12=1/2 (Since we can draw 2 cards from 4 cards in 4*3=12 ways which is total no. of possible cases)