To determine the probability of drawing a king and a queen when selecting two cards at random from a standard deck of 52 cards, it's helpful to break down the process step-by-step.
1. Identify the number of ways to choose 1 king from the 4 kings in the deck.
- There are 4 kings in the deck, and we need to choose 1 of them.
- This can be represented by [tex]\( \binom{4}{1} \)[/tex].
- The value is 4 since [tex]\(\binom{4}{1} = 4\)[/tex].
2. Identify the number of ways to choose 1 queen from the 4 queens in the deck.
- Similarly, there are 4 queens in the deck, and we need to choose 1 of them.
- This can be represented by [tex]\( \binom{4}{1} \)[/tex].
- The value is 4 since [tex]\(\binom{4}{1} = 4\)[/tex].
3. Identify the total number of ways to choose any 2 cards from the 52 cards in the deck.
- There are 52 cards in the deck and we need to choose 2 of them.
- This can be represented by [tex]\( \binom{52}{2} \)[/tex].
- The value is 1326 since [tex]\(\binom{52}{2} = 1326\)[/tex].
4. Calculate the probability of drawing a king and a queen.
- The probability is given by the ratio of the number of favorable outcomes to the total number of outcomes.
- The favorable outcomes consist of selecting 1 king and 1 queen, which equals [tex]\( \binom{4}{1} \times \binom{4}{1} = 4 \times 4 = 16 \)[/tex].
- The total number of possible outcomes of choosing 2 cards from 52 is [tex]\( \binom{52}{2} = 1326 \)[/tex].
- Therefore, the probability [tex]\( P \)[/tex] is:
[tex]\[
P = \frac{\binom{4}{1} \times \binom{4}{1}}{\binom{52}{2}} = \frac{16}{1326} \approx 0.01207
\][/tex]
So, the correct expression that represents the probability of drawing a king and a queen is:
[tex]\[ \frac{\left( \binom{4}{1} \right) \left( \binom{4}{1} \right)}{\binom{52}{2}} \][/tex]
Therefore, the correct option is:
[tex]\[ \frac{\left( { }_4 C_1 \right) \left( { }_4 C_1 \right)}{{ }_{52} C_2} \][/tex]
This accurately represents the probability of drawing a king and a queen from a standard deck of 52 playing cards.
