Question

Here is a table showing all

52

cards in a standard deck.

												Face cards
Color	Suit	Ace	Two	Three	Four	Five	Six	Seven	Eight	Nine	Ten	Jack	Queen	King
Red	Hearts	A ♥	2 ♥	3 ♥	4 ♥	5 ♥	6 ♥	7 ♥	8 ♥	9 ♥	10 ♥	J ♥	Q ♥	K ♥
Red	Diamonds	A ♦	2 ♦	3 ♦	4 ♦	5 ♦	6 ♦	7 ♦	8 ♦	9 ♦	10 ♦	J ♦	Q ♦	K ♦
Black	Spades	A ♠	2 ♠	3 ♠	4 ♠	5 ♠	6 ♠	7 ♠	8 ♠	9 ♠	10 ♠	J ♠	Q ♠	K ♠
Black	Clubs	A ♣	2 ♣	3 ♣	4 ♣	5 ♣	6 ♣	7 ♣	8 ♣	9 ♣	10 ♣	J ♣	Q ♣	K ♣

A card is drawn at random from a standard deck. That card is not put back in the deck, and a second card is drawn at random from the remaining cards in the deck.

What is the probability that both of the cards are hearts?
Do not round your intermediate computations. Round your final answer to four decimal places.

Answer 1

A card is drawn at random from a standard deck. That card is not put back in the deck( i.e. without replacement),and a second card is drawn at random from remaining cards in the deck.

What is the probability that both cards drawn (without replacement) will be Heart?

P( Both are Hearts) =P ( First card is a Heart) *P ( Second card is a Heart)

There are 13 Hearts cards and a card selected at random 52 cards.

P ( First card is a Heart) =

There are 12 Hearts cards left in the deck if one is pulled and not replaced, and 51 total cards remaining.

P ( Second card is a Hearts)=

P ( Both cards are Hearts)=

P ( Both cards are Hearts) =0.05882

2) Alternative Method

There are 52 cards in standard deck and 13 Hearts cards in the deck .

If two cards drawn without replacement from the standard deck of 52 cards. The possible number of ways is:

n= ⁵²C₂

We here find out the probability of both the cards is Hearts .There are the 13 Hearts cards therefore 2 cards selected without replacement from the 13 cards is:

m = ¹³C₂

P ( Both cards are Hearts)

=

= ¹³C₂ / ⁵²C₂

= 0.05882

The probability of both the cards is Hearts is 0.05882.