To determine \( P(A, D) \) when selecting two cards randomly with replacement from a set of 3 cards spelling the word "ADD," we first note the composition of the cards:
- 1 card with 'A'
- 2 cards with 'D'
This gives us a total of 3 cards.
Step 1: Determine Probabilities of Selecting Each Card
- The probability of selecting 'A' is: \[ P(A) = \frac{1}{3} \]
- The probability of selecting 'D' is: \[ P(D) = \frac{2}{3} \]
Step 2: Consider All Possible Outcomes When Selecting 2 Cards with Replacement
Since the selections are made with replacement, each selection is independent. The following pairs of cards can be selected:
- \( (A, A) \)
- \( (A, D) \)
- \( (D, A) \)
- \( (D, D) \)
Given the independence of the card selections, the probabilities for these outcomes can be calculated:
- \( P(A, A) = P(A) \cdot P(A) = \frac{1}{3} \cdot \frac{1}{3} = \frac{1}{9} \)
- \( P(A, D) = P(A) \cdot P(D) = \frac{1}{3} \cdot \frac{2}{3} = \frac{2}{9} \)
- \( P(D, A) = P(D) \cdot P(A) = \frac{2}{3} \cdot \frac{1}{3} = \frac{2}{9} \)
- \( P(D, D) = P(D) \cdot P(D) = \frac{2}{3} \cdot \frac{2}{3} = \frac{4}{9} \)
Step 3: Calculate \( P(A, D) \)
Now, we need to find the total probability of the event "getting one A and one D" in two selections:
- The event can happen in two ways: either \( (A, D) \) or \( (D, A) \).
Thus, the total probability \( P(A, D) \) is: \[ P(A, D) = P(A, D) + P(D, A) = \frac{2}{9} + \frac{2}{9} = \frac{4}{9} \]
Final Result
Therefore, the probability \( P(A, D) \) when selecting two cards with replacement is:
\[ \boxed{\frac{4}{9}} \]