Sure, let’s find the probability step-by-step!
### Problem Statement:
We need to find the probability of drawing three queens from a standard deck of 52 cards, given that the first card drawn was a queen. The cards are not replaced.
### Steps to Solution:
1. Understanding the Setup:
- A standard deck has 52 cards.
- There are 4 queens in the deck.
- The first card drawn is a queen.
2. First Draw:
- Since the first card drawn is a queen, it’s a given condition.
3. Second Draw:
- After drawing the first queen, there are now 3 queens left in a deck of 51 remaining cards.
- Probability of drawing a queen on the second draw is:
[tex]\[
\text{Probability} = \frac{\text{Number of remaining queens}}{\text{Number of remaining cards}} = \frac{3}{51} \approx 0.0588
\][/tex]
4. Third Draw:
- After successfully drawing a second queen, there are now 2 queens left in a deck of 50 remaining cards.
- Probability of drawing a queen on the third draw is:
[tex]\[
\text{Probability} = \frac{\text{Number of remaining queens}}{\text{Number of remaining cards}} = \frac{2}{50} = 0.04
\][/tex]
5. Combined Probability:
- To find the overall probability of these two events happening consecutively (drawing a queen on the second and then on the third draw), we multiply the probabilities of each independent event:
[tex]\[
\text{Total Probability} = \left(\frac{3}{51}\right) \times \left(\frac{2}{50}\right) = 0.058823529411764705 \times 0.04 = 0.0023529411764705885
\][/tex]
6. Simplifying the Fraction:
- We already found that the numerical value of the total probability is approximately 0.0023529411764705885.
- This value corresponds to the fraction [tex]\(\frac{1}{425}\)[/tex] in its simplest form.
Thus, the final answer is:
[tex]\[
\text{Probability} = \frac{1}{425}
\][/tex]
Therefore, the probability of drawing three queens in a row (with the first being given) from a standard deck of cards, without replacement, is: [tex]\(\frac{1}{425}\)[/tex].
