To determine the probability [tex]\( P(A, D) \)[/tex] where two cards are drawn with replacement from a set that spells out the word "ADD," we can follow these detailed steps:
1. Identify the Card Set and Probabilities:
- The set of cards includes: A, D, D.
- The total number of cards is 3.
2. Define the Probabilities for Each Event:
- Probability of drawing an 'A' (P(A)): Since there is only 1 'A' card out of 3 cards:
[tex]\[
P(A) = \frac{1}{3}
\][/tex]
- Probability of drawing a 'D' (P(D)): Since there are 2 'D' cards out of 3 cards:
[tex]\[
P(D) = \frac{2}{3}
\][/tex]
3. Calculate the Probability of Drawing 'A' first and then 'D':
[tex]\[
P(A \text{ then } D) = P(A) \times P(D) = \frac{1}{3} \times \frac{2}{3} = \frac{2}{9}
\][/tex]
4. Calculate the Probability of Drawing 'D' first and then 'A':
[tex]\[
P(D \text{ then } A) = P(D) \times P(A) = \frac{2}{3} \times \frac{1}{3} = \frac{2}{9}
\][/tex]
5. Calculate the Combined Probability for Either Scenario:
Since [tex]\( P(A, D) \)[/tex] includes both scenarios where 'D' is the second card (either drawing 'A' first followed by 'D' or drawing 'D' first followed by 'A'):
[tex]\[
P(A, D) = P(A \text{ then } D) + P(D \text{ then } A) = \frac{2}{9} + \frac{2}{9} = \frac{4}{9}
\][/tex]
Thus, the probability [tex]\( P(A, D) \)[/tex] is [tex]\(\frac{4}{9}\)[/tex].
Therefore, the correct answer is [tex]\(\frac{4}{9}\)[/tex].
