Question

There are two 7s (of hearts and spades) and two 8s (of hearts and spades) in a deck of cards. The deck has no other cards. Emma draws two cards from this deck and says “I have a 7”. What is the probability of Emma having both 7s? What if she says “I have a 7 of hearts”?

Answer 1

There are two 7s (of spaded and hearts) and two 8s (of spade and heart) in deck of cards and not other cards are there.

Total number of cards in deck = two 7s + two 8s = 4

2 cards are drawn from the deck.

Probability of Emma having both 7s that is P(both 7s) = P(First is 7) * P(Second is 7)

$P(first \ is \ 7) = \frac{number \ of \ 7s}{total \ number \ of \ cards \ in \ deck} = \frac{2}{4}$

One 7 is already selected, so only one 7 is remain and total 3 are there

$P(first \ is \ 7) = \frac{number \ of \ remaining \ 7s}{total \ remaining \ cards \ in \ deck} = \frac{1}{3}$

P(Emma having both 7s) = 2/4 * 1/3 = 2/12 = 1/6

P(both 7s) = 1/6

If she says "I have a 7 of hearts" then

There is only one 7 card which is of heart in total 4

$P(7 \ of \ hearts) = \frac{number \ of \ 7 \ hearts \ card}{toral \ cards} = \frac{1}{4}$