Question

b. drawing a red card from a deck of cards, not replacing it, and then drawing a queen from the remaining cards.

Explanation:

To determine if the two events (drawing a red card and then drawing a queen without replacement) are dependent or to find the probability, we analyze:

Step 1: Probability of first event (drawing a red card)

A standard deck has 52 cards, 26 red.
Probability: \( P(\text{red}) = \frac{26}{52} = \frac{1}{2} \).

Step 2: Probability of second event (drawing a queen)

• If the first card was a red queen (2 red queens), remaining cards: 51, queens left: 3.

\( P(\text{queen}|\text{red queen}) = \frac{3}{51} \).

• If the first card was a red non - queen (24 red non - queens), remaining cards: 51, queens left: 4.

\( P(\text{queen}|\text{red non - queen}) = \frac{4}{51} \).

Since the outcome of the first event affects the second, the events are dependent.

To find the overall probability of both events:
Case 1: First card is red queen (probability \( \frac{2}{52} \)), then queen:
\( P(\text{red queen then queen})=\frac{2}{52}\times\frac{3}{51}=\frac{6}{2652} \)

Case 2: First card is red non - queen (probability \( \frac{24}{52} \)), then queen:
\( P(\text{red non - queen then queen})=\frac{24}{52}\times\frac{4}{51}=\frac{96}{2652} \)

Total probability: \( \frac{6 + 96}{2652}=\frac{102}{2652}=\frac{1}{26}\approx0.0385 \)

If the question was about dependence:

The first event (drawing red) changes the deck composition (number of queens left), so the probability of drawing a queen depends on the first draw.

Answer:

The two events are dependent.

If the question was about probability: