Two cards are drawn from a standard deck of cards.

Question

Two cards are drawn from a standard deck of cards. Find the probability that the cards are both face cards or both diamonds. (Assume that face cards include Kings, Queens, and Jacks)

A. 24/442
B. 47/442
C. 46/221
D. 47/221

Answer 1

There are 52 cards in the deck. 13 are diamonds. There are also 9 other face cards that are not diamonds. Therefore the probability of drawing either a face card or a diamond is 22/52 on the first draw. this card is not replaced. The probability of drawing either a face or a diamond is now 21/51. The probability of doing both of these is their product.

 

22/52 * 21/51 = 462/2652 which can be reduced to 77/442.

 

I find it very strange that this is not one of the choices listed in the 4 possible responses.

Answer 2

A deck of cards has 52 cards of which

4+ 4+4=12 face cards and

13 cards are diamonds.

 

Let' list the probabilities separately.

The probability of the first card being a face card and the probability of the second card being also a face card:

P(F₁)=12/52 and P(F₂)=11/51.

Therefore,

P(F₁and F₂)=(12/52)(11/51)

 

The probability of the first and the second cards being diamond cards is given by,

P(D₁) =13/52 and P(D₂) =12/51.

Therefore,

P(D₁ and D₂)=(13/52)(12/51)

 

The probability that the cards drawn are both face card and diamond will be,

(3/52)(2/51)

 

Now add the prob. face cards and the diamond cards, because the question says OR, then subtract the prob. when these two events occurs at the same time.(to eliminate repetition)

 

P=(12/52)(11/51)+(13/52)(12/51)-(3/52)(2/51)

P=47/442

choice b. is the correct response