Probability of certain cards in a 48 card pinochle deck (please show work)

Question

A card is drawn randomly from a standard 48-card pinochle deck. Find the probability of drawing the given card. (Note that a pinochle deck consists of all four suits. The cards 9, 10, jack, queen, king, ace appear twice in each suit. There are no 2, 3, 4, 5, 6, 7, or 8s.)



6. any ace

7. any black queen

8. any heart

9. any 9 or 10

10. any ace of hearts

11. any 7 

(Please show work!)

Thanks!

Answer 1

Probability=(What you want)/(What you got)

In all cases, "what you got" is 48 because there are 48 cards in the deck. Just figure out how many of the wanted card there are in the deck and divide that number by 48.

(6) There are 8 aces--2 from each suit. 8/48=1/6

(7) There are 4 black queens--2 spades and 2 clubs. 4/48=1/12

(8) There are 12 hearts--2 9s, 210s, 2 jacks, 2 queens, 2 kings, and 2 aces. 12/48=1/4

(9) There are 8 9s and 8 10s, or a total of 16 9s and 10s. 16/48=1/3

(10) There are 2 aces of hearts. 2/48=1/24

(11) Same as 6.

Answer 2

Hi Ty,

In this case, for each of the given situations, we're drawing only one card, so the probability that we draw a particular type of card will be equal to the (# of possible cards that meet our desired requirement)/(# of total cards).

In the description of the pinochle deck we're using, it says there are two of each card, 9 through Ace, in each suit - so two 9 of clubs cards, two 9 of diamonds cards, etc. There are 48 cards total, so our denominator for each possibility will always be 48 - and then we will simplify the fractions the best we can.

6) The probability of drawing any ace would be (# of aces)/(# of total cards). Since there are two of every card 9 through Ace, we know there are 8 aces. So our probability is (8 aces)/(48 cards) = 1/6 chance of drawing any ace.

7) How many black queens are there? 2 queen of spade cards, and 2 queen of club cards, so (4 black queens)/(48 cards) = 1/12 chance of drawing a black queen.

8) How many heart cards are there? 2 each of 9-Ace makes (12 Heart cards)/(48 cards) = 1/4 chance of drawing a heart card.

9) With two of each card, there are 8 9's and 8 10's, so (16 9's/10's)/(48 cards) = 1/3 chance of drawing a 9 or 10.

10) How many Ace of hearts are in the deck? (2 Ace of hearts)/(48 cards) = 1/24 chance of drawing an ace of hearts.

11) Since the deck only includes 9 through Ace, there are no 7's in the deck. (0 7's)/(48 cards) = 0 chance of drawing any 7 cards.

I hope this helps!

Andy