A pack of cards consists of 15 cards numbered 1 to 15. Three cards are drawn at random with replacement. Then, the probability of getting 2odd and one even numbered cards is

Options

A. `(348)/(1125)`
B. `(398)/(1125)`
C. `(448)/(1125)`
D. `(498)/(1125)`

Explanation

To solve the problem of finding the probability of drawing 2 odd and 1 even numbered card from a pack of 15 cards (numbered 1 to 15) with replacement, we can follow these steps:

### Step 1: Identify the total number of cards and their types
- The pack contains 15 cards numbered from 1 to 15.
- Odd numbered cards: 1, 3, 5, 7, 9, 11, 13, 15 (Total: 8 odd cards)
- Even numbered cards: 2, 4, 6, 8, 10, 12, 14 (Total: 7 even cards)

### Step 2: Determine the probabilities of drawing odd and even cards
- Probability of drawing an odd card (P(Odd)):
\[ P(Odd) = \frac{\text{Number of odd cards}}{\text{Total number of cards}} = \frac{8}{15} \]
- Probability of drawing an even card (P(Even)):
\[ P(Even) = \frac{\text{Number of even cards}}{\text{Total number of cards}} = \frac{7}{15} \]

### Step 3: Identify the different arrangements for drawing 2 odd and 1 even card
The different arrangements for drawing 2 odd and 1 even card can be:
1. Odd, Odd, Even (OOE)
2. Odd, Even, Odd (OEO)
3. Even, Odd, Odd (EOO)

### Step 4: Calculate the probability for each arrangement
- For the arrangement OOE:
\[ P(OOE) = P(Odd) \times P(Odd) \times P(Even) = \left(\frac{8}{15}\right) \times \left(\frac{8}{15}\right) \times \left(\frac{7}{15}\right) = \frac{8 \times 8 \times 7}{15 \times 15 \times 15} = \frac{448}{3375} \]

- For the arrangement OEO:
\[ P(OEO) = P(Odd) \times P(Even) \times P(Odd) = \left(\frac{8}{15}\right) \times \left(\frac{7}{15}\right) \times \left(\frac{8}{15}\right) = \frac{448}{3375} \]

- For the arrangement EOO:
\[ P(EOO) = P(Even) \times P(Odd) \times P(Odd) = \left(\frac{7}{15}\right) \times \left(\frac{8}{15}\right) \times \left(\frac{8}{15}\right) = \frac{448}{3375} \]

### Step 5: Sum the probabilities of all arrangements
Now, we add the probabilities of all three arrangements:
\[ P(2 \text{ Odd and } 1 \text{ Even}) = P(OOE) + P(OEO) + P(EOO) = \frac{448}{3375} + \frac{448}{3375} + \frac{448}{3375} = 3 \times \frac{448}{3375} = \frac{1344}{3375} \]

### Final Answer
Thus, the probability of getting 2 odd and 1 even numbered card when drawing 3 cards with replacement is:
\[ \frac{1344}{3375} \]