To determine the probability [tex]\( P(A, D) \)[/tex] of drawing an 'A' first and a 'D' second when two cards are drawn with replacement, let's analyze the problem step-by-step.
1. Identify the individual probabilities:
- The set of cards consists of the letters A, D, and D.
- There are 3 cards in total.
Step 1: Calculate the probability of drawing an 'A' (denoted as [tex]\( P(A) \)[/tex]):
- There is 1 'A' card out of 3 cards.
[tex]\[ P(A) = \frac{1}{3} \][/tex]
Step 2: Calculate the probability of drawing a 'D' (denoted as [tex]\( P(D) \)[/tex]):
- There are 2 'D' cards out of 3 cards.
[tex]\[ P(D) = \frac{2}{3} \][/tex]
2. Since the cards are replaced, these events are independent:
- This means the probability of drawing 'A' first and then drawing 'D' can be found by multiplying the individual probabilities.
3. Calculate the combined probability [tex]\( P(A, D) \)[/tex]:
- The probability of drawing 'A' first and then 'D' second is:
[tex]\[
P(A \text{ then } D) = P(A) \times P(D)
\][/tex]
- Now, substitute the individual probabilities we calculated:
[tex]\[
P(A \text{ then } D) = \frac{1}{3} \times \frac{2}{3} = \frac{1 \times 2}{3 \times 3} = \frac{2}{9}
\][/tex]
Therefore, the probability [tex]\( P(A, D) \)[/tex] of drawing 'A' first and 'D' second with replacement is [tex]\( \frac{2}{9} \)[/tex].
So, the correct answer is [tex]\( \frac{2}{9} \)[/tex].
