Question
You randomly draw two cards from a standard deck. Suppose you want to draw a queen twice in a row. What is the probability of picking a queen on the second draw if the events are dependent?(1 point) Responses 452 Start Fraction 4 over 52 End Fraction 452⋅452 Start Fraction 4 over 52 End Fraction times Start Fraction 4 over 52 End Fraction 452⋅351 Start Fraction 4 over 52 End Fraction times Start Fraction 3 over 51 End Fraction 351
Answers
GPT-4o mini
To find the probability of drawing a queen on the second draw after drawing a queen on the first draw (where the events are dependent), we can analyze it step by step.
1. On the first draw, the probability of drawing a queen:
\[
P(\text{Queen on 1st draw}) = \frac{4}{52}
\]
2. If a queen is drawn on the first draw, there are now 51 cards left in the deck, and only 3 queens remaining. Therefore, the probability of drawing a queen on the second draw, given that the first draw was a queen, is:
\[
P(\text{Queen on 2nd draw | Queen on 1st draw}) = \frac{3}{51}
\]
3. To find the combined probability of both events happening (drawing a queen on the first draw and then drawing a queen on the second draw), we multiply these two probabilities:
\[
P(\text{Queen on 1st and Queen on 2nd}) = P(\text{Queen on 1st draw}) \times P(\text{Queen on 2nd draw | Queen on 1st draw}) = \frac{4}{52} \times \frac{3}{51}
\]
However, since you've asked specifically for the probability of drawing a queen on the second draw after having already drawn a queen, we only need the second part:
\[
P(\text{Queen on 2nd draw | Queen on 1st draw}) = \frac{3}{51}
\]
Thus, the probability of picking a queen on the second draw (given that the first was also a queen) is:
\[
\boxed{\frac{3}{51}}
\]
1. On the first draw, the probability of drawing a queen:
\[
P(\text{Queen on 1st draw}) = \frac{4}{52}
\]
2. If a queen is drawn on the first draw, there are now 51 cards left in the deck, and only 3 queens remaining. Therefore, the probability of drawing a queen on the second draw, given that the first draw was a queen, is:
\[
P(\text{Queen on 2nd draw | Queen on 1st draw}) = \frac{3}{51}
\]
3. To find the combined probability of both events happening (drawing a queen on the first draw and then drawing a queen on the second draw), we multiply these two probabilities:
\[
P(\text{Queen on 1st and Queen on 2nd}) = P(\text{Queen on 1st draw}) \times P(\text{Queen on 2nd draw | Queen on 1st draw}) = \frac{4}{52} \times \frac{3}{51}
\]
However, since you've asked specifically for the probability of drawing a queen on the second draw after having already drawn a queen, we only need the second part:
\[
P(\text{Queen on 2nd draw | Queen on 1st draw}) = \frac{3}{51}
\]
Thus, the probability of picking a queen on the second draw (given that the first was also a queen) is:
\[
\boxed{\frac{3}{51}}
\]