Answer :
To find the probability of drawing one king and one queen when two cards are chosen at random from a standard deck of 52 cards, we can follow these steps:
1. Identify the total number of ways to draw 2 cards from a deck of 52 cards:
- The number of ways to choose 2 cards out of 52 is given by the combination formula [tex]\( \binom{52}{2} \)[/tex], which is calculated as:
[tex]\[ \binom{52}{2} = \frac{52!}{2!(52-2)!} = \frac{52!}{2! \cdot 50!} = \frac{52 \cdot 51}{2 \cdot 1} = 1326 \][/tex]
2. Identify the number of ways to draw one king and one queen:
- There are 4 kings in the deck and 4 queens in the deck.
- The number of ways to choose one king out of 4 is [tex]\( \binom{4}{1} = 4 \)[/tex]
- The number of ways to choose one queen out of 4 is [tex]\( \binom{4}{1} = 4 \)[/tex]
- Therefore, the number of ways to draw one king and one queen is [tex]\( 4 \times 4 = 16 \)[/tex]
3. Calculate the probability:
- We divide the number of favorable outcomes by the total number of outcomes:
[tex]\[ \text{Probability} = \frac{\text{Number of ways to draw one king and one queen}}{\text{Total number of ways to draw 2 cards from 52}} \][/tex]
[tex]\[ \text{Probability} = \frac{16}{1326} = \frac{8}{663} \approx 0.0121 \][/tex]
Among the given expressions, let's find the one that matches our steps:
- The correct expression would provide the ratio of the product of the combinations of drawing one king and one queen over the combination of drawing two cards from the deck.
Considering each option:
1. [tex]\(\frac{\left(4 P_1\right)\left({ }_3 F_1\right)}{52 P_2}\)[/tex]: Incorrect as it uses permutations and an unknown notation.
2. [tex]\(\frac{\left(C_1\right)\left(3 C_1\right)}{52 C_2}\)[/tex]: Incorrect as the notation does not match our goal.
3. [tex]\(\frac{\left(4 P_1\right)\left(4 P_1\right)}{52 P_2}\)[/tex]: Incorrect as it uses permutations instead of combinations.
4. [tex]\(\frac{\left.6_1 C_1\right)\left(C_1\right)}{12 C_2}\)[/tex]: Incorrect as the numbers and operations do not match our steps.
Although none of these options directly match our calculation:
The comprehensive detailed solution derived above, using our own calculation steps, yields the correct calculation and understanding of the probability of drawing one king and one queen in a standard deck of 52 cards.
1. Identify the total number of ways to draw 2 cards from a deck of 52 cards:
- The number of ways to choose 2 cards out of 52 is given by the combination formula [tex]\( \binom{52}{2} \)[/tex], which is calculated as:
[tex]\[ \binom{52}{2} = \frac{52!}{2!(52-2)!} = \frac{52!}{2! \cdot 50!} = \frac{52 \cdot 51}{2 \cdot 1} = 1326 \][/tex]
2. Identify the number of ways to draw one king and one queen:
- There are 4 kings in the deck and 4 queens in the deck.
- The number of ways to choose one king out of 4 is [tex]\( \binom{4}{1} = 4 \)[/tex]
- The number of ways to choose one queen out of 4 is [tex]\( \binom{4}{1} = 4 \)[/tex]
- Therefore, the number of ways to draw one king and one queen is [tex]\( 4 \times 4 = 16 \)[/tex]
3. Calculate the probability:
- We divide the number of favorable outcomes by the total number of outcomes:
[tex]\[ \text{Probability} = \frac{\text{Number of ways to draw one king and one queen}}{\text{Total number of ways to draw 2 cards from 52}} \][/tex]
[tex]\[ \text{Probability} = \frac{16}{1326} = \frac{8}{663} \approx 0.0121 \][/tex]
Among the given expressions, let's find the one that matches our steps:
- The correct expression would provide the ratio of the product of the combinations of drawing one king and one queen over the combination of drawing two cards from the deck.
Considering each option:
1. [tex]\(\frac{\left(4 P_1\right)\left({ }_3 F_1\right)}{52 P_2}\)[/tex]: Incorrect as it uses permutations and an unknown notation.
2. [tex]\(\frac{\left(C_1\right)\left(3 C_1\right)}{52 C_2}\)[/tex]: Incorrect as the notation does not match our goal.
3. [tex]\(\frac{\left(4 P_1\right)\left(4 P_1\right)}{52 P_2}\)[/tex]: Incorrect as it uses permutations instead of combinations.
4. [tex]\(\frac{\left.6_1 C_1\right)\left(C_1\right)}{12 C_2}\)[/tex]: Incorrect as the numbers and operations do not match our steps.
Although none of these options directly match our calculation:
The comprehensive detailed solution derived above, using our own calculation steps, yields the correct calculation and understanding of the probability of drawing one king and one queen in a standard deck of 52 cards.
