If you are dealt 4 cards from a shuffled deck of 52 cards, find the probability of getting three queens and one king.
To find the probability of getting three queens and one king, we first need to determine the total number of possible sets of 4 cards that can be dealt from a shuffled deck of 52 cards.
The total number of possible sets of 4 cards from a deck of 52 is given by the combination formula:
C(n, r) = n! / (r!(n - r)!), where n is the total number of items and r is the number of items to be chosen.
In this case, n = 52 (total number of cards in the deck) and r = 4 (number of cards being dealt).
So, the total number of possible sets of 4 cards is:
C(52, 4) = 52! / (4!(52 - 4)!) = 52! / (4!48!) = (52 * 51 * 50 * 49) / (4 * 3 * 2 * 1) = 270,725.
Now, we need to determine the number of sets that contain three queens and one king. There are 4 queens and 4 kings in the deck.
To choose 3 queens out of 4, we use the combination formula:
C(4, 3) = 4! / (3!(4 - 3)!) = 4! / (3!1!) = 4 / 1 = 4.
To choose 1 king out of 4, we use the combination formula:
C(4, 1) = 4! / (1!(4 - 1)!) = 4! / (1!3!) = 4 / 1 = 4.
To get the number of sets with three queens and one king, we multiply the number of ways to choose the queens by the number of ways to choose the king:
4 * 4 = 16.
Therefore, the number of sets with three queens and one king is 16.
So, the probability of getting three queens and one king is calculated by dividing the number of sets with three queens and one king by the total number of possible sets of 4 cards:
P(three queens and one king) = Number of sets with three queens and one king / Total number of possible sets of 4 cards
= 16 / 270,725
≈ 0.0000591
Therefore, the probability of getting three queens and one king when dealt 4 cards from a shuffled deck of 52 cards is approximately 0.0000591, or about 0.00591%.