You select 9 cards randomly from a deck of 52 cards.
What is the probability that all of the cards selected are face cards (i.e. jacks, queens, or kings)?
To calculate the probability of selecting all face cards from a deck of 52 cards, we need to determine the total number of possible outcomes and the number of successful outcomes.
First, let's determine the total number of possible outcomes. We are selecting 9 cards randomly from a deck of 52 cards, so the total number of possible outcomes can be calculated using the combination formula:
C(n, r) = n! / (r!(n-r)!),
where n is the total number of cards (52) and r is the number of cards selected (9).
Therefore, the total number of possible outcomes is C(52, 9) = 52! / (9!(52-9)!) = 752,538,150.
Next, let's determine the number of successful outcomes, which in this case are the 9 face cards (Jacks, Queens, or Kings). In a standard deck, there are 4 face cards of each suit (Jacks of Hearts, Diamonds, Clubs, and Spades, Queens of Hearts, Diamonds, Clubs, and Spades, and Kings of Hearts, Diamonds, Clubs, and Spades). So, there are a total of 4 * 3 = 12 face cards in the deck.
Since we need to select all 9 face cards, the number of successful outcomes is C(12, 9) = 12! / (9!(12-9)!) = 220.
Finally, we can calculate the probability by dividing the number of successful outcomes by the total number of possible outcomes:
Probability = Number of successful outcomes / Total number of possible outcomes = 220 / 752,538,150.
Therefore, the probability that all 9 cards selected are face cards is approximately 0.000000292 (or about 2.92 in 10 million).