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 two odd and one even numbered cards, is?

Options

A. a. `3/1430`
B. b. `100/2430`
C. c. `448/1125`
D. d. `7/72`

Answer

To solve the problem of finding the probability of drawing two odd and one even numbered card from a pack of 15 cards numbered 1 to 15, we can follow these steps:

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

### Step 2: Determine the total number of cards
- Total cards = 15 (8 odd + 7 even)

### Step 3: Calculate the probability of drawing two odd and one even card
Since the draws are with replacement, the probability remains constant for each draw.

- Probability of drawing an odd card (P(Odd)) = Number of odd cards / Total cards = 8/15
- Probability of drawing an even card (P(Even)) = Number of even cards / Total cards = 7/15

### Step 4: Determine the different arrangements of drawing two odd and one even card
The possible arrangements for drawing two odd and one even card can be represented as:
- OOE, OEO, EOO

### Step 5: Calculate the probability for one arrangement
For one arrangement (e.g., OOE):
- Probability = P(Odd) × P(Odd) × P(Even)
- Probability = (8/15) × (8/15) × (7/15)

### Step 6: Calculate the total probability for all arrangements
Since there are 3 arrangements (OOE, OEO, EOO), we multiply the probability of one arrangement by the number of arrangements:
- Total Probability = 3 × [(8/15) × (8/15) × (7/15)]

### Step 7: Simplify the expression
- Total Probability = 3 × [(64/225) × (7/15)]
- Total Probability = 3 × (448/3375)
- Total Probability = 1344/3375

### Step 8: Final Result
The probability of getting two odd and one even numbered card is:
- **Probability = 1344/3375**