Questions: Let two cards be dealt successively, without replacement, from a standard 52-card deck. Find the probability of the event. two queens The probability of drawing two queens is (Simplify your answer. Type an integer or a fraction.)

Solution Steps

To find the probability of drawing two queens successively without replacement from a standard 52-card deck, we need to consider the following steps:

1. Calculate the probability of drawing a queen on the first draw.
2. Calculate the probability of drawing a queen on the second draw, given that a queen was already drawn on the first draw.
3. Multiply these two probabilities to get the final probability.

The probability of drawing a queen on the first draw from a standard 52-card deck is given by:

\[ P(\text{First Queen}) = \frac{\text{Number of Queens}}{\text{Total Cards}} = \frac{4}{52} = \frac{1}{13} \approx 0.0769 \]

After drawing one queen, there are now 3 queens left in a total of 51 cards. The probability of drawing a queen on the second draw is:

\[ P(\text{Second Queen} | \text{First Queen}) = \frac{\text{Remaining Queens}}{\text{Remaining Cards}} = \frac{3}{51} = \frac{1}{17} \approx 0.0588 \]

The total probability of drawing two queens successively without replacement is the product of the probabilities from the first and second draws:

\[ P(\text{Two Queens}) = P(\text{First Queen}) \times P(\text{Second Queen} | \text{First Queen}) = \frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221} \approx 0.0045 \]

Final Answer