Selecting Two Queens from a Deck

Solution & Explanation

To find the probability of drawing two queens consecutively from a standard deck of 52 cards, we need to consider both the probability of drawing a queen on the first draw and the probability of drawing another queen on the second draw, given that the first card drawn was a queen.

Step-by-Step Calculation:

1. Probability of Drawing the First Queen:

   • A standard deck of cards contains 52 cards, among which there are 4 queens.
   • The probability of drawing a queen on the first draw is: $P(\text{First Queen}) = \frac{4}{52} = \frac{1}{13}$

2. Probability of Drawing the Second Queen:

   • After drawing the first queen, there are now 51 cards left in the deck, with 3 queens remaining.
   • The probability of drawing a second queen is: $P(\text{Second Queen | First Queen}) = \frac{3}{51} = \frac{1}{17}$

3. Probability of Both Events Occurring:

   • Since drawing the first and second queen are dependent events (the outcome of the first affects the second), we find the joint probability by multiplying the probabilities of each event: $P(\text{Two Queens}) = P(\text{First Queen}) \times P(\text{Second Queen | First Queen})$ $P(\text{Two Queens}) = \frac{1}{13} \times \frac{1}{17} = \frac{1}{221}$

Alternative Method Using Combinations:

We can also solve this problem using combinations to find the number of favorable outcomes over the total possible outcomes:

1. Combinations of Drawing 2 Queens from 4 Queens:

   • The number of ways to choose 2 queens from 4 queens is given by the combination formula: $\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6$

2. Combinations of Drawing Any 2 Cards from 52 Cards:

   • The number of ways to choose any 2 cards from 52 cards is: $\binom{52}{2} = \frac{52 \times 51}{2 \times 1} = 1326$

3. Probability Using Combinations:

   • The probability of drawing 2 queens is the ratio of the number of favorable outcomes to the total outcomes: $P(\text{Two Queens}) = \frac{\binom{4}{2}}{\binom{52}{2}} = \frac{6}{1326} = \frac{1}{221}$

Conclusion:

Both methods confirm that the probability of drawing two queens consecutively from a standard deck of cards is $\frac{1}{221}$. This probability reflects the rare occurrence of such an event, given the distribution of queens within the deck.