A standard deck of 52 playing cards contains four of each numbered card 2-10 and four each of aces, kings, queens, and jacks. Two cards are chosen from the deck at random.
Which expression represents the probability of drawing a king and a queen?
[tex]$\frac{\left({ }_4 P_1\right)\left({ }_3 P_1\right)}{{ }_{52} P_2}$[/tex]
[tex]$\frac{\left({ }_4 C_1\right)\left({ }_3 C_1\right)}{{ }_{52} C_2}$[/tex]
[tex]$\frac{\left({ }_4 P_1\right)\left({ }_4 P_1\right)}{52 P_2}$[/tex]
[tex]$\frac{\left({ }_4 C_1\right)\left(C_4 C_1\right)}{{ }_{52} C_2}$[/tex]
Answer (1)
Calculate the number of ways to choose a king: 4 C 1 = 4 .
Calculate the number of ways to choose a queen: 4 C 1 = 4 .
Calculate the total number of ways to choose two cards: 52 C 2 = 1326 .
The probability of drawing a king and a queen is 52 C 2 ( 4 C 1 ) ( 4 C 1 ) = 52 C 2 ( 4 C 1 ) ( 4 C 1 ) .
Explanation
Understand the problem We are given a standard deck of 52 playing cards, which contains four of each numbered card 2-10 and four each of aces, kings, queens, and jacks. We want to find the probability of drawing a king and a queen when two cards are chosen at random.
Outline the solution To solve this, we need to determine the number of ways to choose one king and one queen, and divide it by the total number of ways to choose two cards from the deck. Since the order in which we draw the cards doesn't matter, we will use combinations.
Calculate the number of ways to choose a king The number of ways to choose one king from the four kings is given by the combination formula n C r = r ! ( n − r )! n ! , where n is the total number of items and r is the number of items to choose. In this case, we have 4 C 1 = 1 ! ( 4 − 1 )! 4 ! = 1 ! 3 ! 4 ! = ( 1 ) ( 3 × 2 × 1 ) 4 × 3 × 2 × 1 = 4 .
Calculate the number of ways to choose a queen Similarly, the number of ways to choose one queen from the four queens is 4 C 1 = 1 ! ( 4 − 1 )! 4 ! = 1 ! 3 ! 4 ! = 4 .
Calculate the total number of ways to choose two cards The total number of ways to choose two cards from the 52 cards in the deck is 52 C 2 = 2 ! ( 52 − 2 )! 52 ! = 2 ! 50 ! 52 ! = 2 × 1 52 × 51 = 1326 .
Calculate the number of ways to choose a king and a queen The number of ways to choose one king and one queen is the product of the number of ways to choose a king and the number of ways to choose a queen, which is 4 × 4 = 16 .
Calculate the probability Therefore, the probability of drawing a king and a queen is the number of ways to choose a king and a queen divided by the total number of ways to choose two cards, which is 1326 16 = 663 8 .
Find the matching expression Now, let's compare this to the given options. The expression that represents the probability of drawing a king and a queen is 52 C 2 ( 4 C 1 ) ( 4 C 1 ) .
State the final answer The correct expression is 52 C 2 ( 4 C 1 ) ( 4 C 1 ) .
Examples
Consider a scenario where you're organizing a card game tournament. Knowing the probability of drawing specific card combinations, like a king and a queen, helps you design fair rules and estimate the likelihood of certain hands occurring. This ensures that the game remains balanced and engaging for all players, preventing any single player from having an unfair advantage due to extremely rare or common hands. For instance, if the probability of a particular hand is too high, you might adjust the scoring system to compensate.
