To determine the probability [tex]\( P(D, D) \)[/tex] of drawing two 'D' cards in succession with replacement from a set of three cards spelling the word [tex]\( A D D \)[/tex], follow these steps:
1. Identify Total and Desired Outcomes:
- The total number of cards is 3.
- Two of these cards are 'D' cards.
2. Calculate Probability for First Card:
The probability of drawing a 'D' card on the first draw is:
[tex]\[
P(\text{first 'D'}) = \frac{\text{number of 'D' cards}}{\text{total number of cards}} = \frac{2}{3}
\][/tex]
3. Replacement Scenario:
Since we are drawing with replacement, the total number of cards remains the same for the second draw, and the probabilities do not change.
4. Calculate Probability for Second Card:
The probability of drawing a 'D' card on the second draw also remains:
[tex]\[
P(\text{second 'D'}) = \frac{2}{3}
\][/tex]
5. Calculate Combined Probability:
The combined probability of both events happening (drawing a 'D' card on the first draw and another 'D' card on the second draw) is the product of their individual probabilities:
[tex]\[
P(D, D) = P(\text{first 'D'}) \times P(\text{second 'D'}) = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9}
\][/tex]
Thus, the correct answer is: [tex]\(\frac{4}{9}\)[/tex].
So, [tex]\( P(D, D) = \frac{4}{9} \)[/tex].
