An unbiased cubic die marked with 1,2,2,3,3,3 is rolled 3 times. The probability of getting a total score of 4 or 6 is `16//216` b. `50//216` c. `60//216` d. none of these

Options

A. `16//216`
B. `50//216`
C. `60//216`
D. None of these

Explanation

To solve the problem of finding the probability of getting a total score of 4 or 6 when rolling the die marked with 1, 2, 2, 3, 3, 3 three times, we can follow these steps:

### Step 1: Understand the Die
The die has the following faces: 1, 2, 2, 3, 3, 3.
- Probability of rolling a 1: \( P(1) = \frac{1}{6} \)
- Probability of rolling a 2: \( P(2) = \frac{2}{6} = \frac{1}{3} \)
- Probability of rolling a 3: \( P(3) = \frac{3}{6} = \frac{1}{2} \)

### Step 2: Total Outcomes
When rolling the die 3 times, the total number of outcomes is:
\[ 6^3 = 216 \]

### Step 3: Find Favorable Outcomes for Total Score of 4
To get a total score of 4, the possible combinations of rolls are:
1. \( (1, 1, 2) \)
2. \( (2, 2, 2) \)

**For (1, 1, 2)**:
- The arrangements can be calculated using the formula for permutations of multiset:
\[ \frac{3!}{2!} = 3 \text{ (since there are two 1's)} \]

**For (2, 2, 2)**:
- There is only 1 arrangement.

Thus, the total favorable outcomes for a score of 4:
\[ 3 + 1 = 4 \]

### Step 4: Find Favorable Outcomes for Total Score of 6
To get a total score of 6, the possible combinations of rolls are:
1. \( (1, 2, 3) \)
2. \( (2, 2, 2) \)
3. \( (3, 3, 3) \)

**For (1, 2, 3)**:
- The arrangements can be calculated as:
\[ 3! = 6 \]

**For (2, 2, 2)**:
- There is only 1 arrangement.

**For (3, 3, 3)**:
- There is only 1 arrangement.

Thus, the total favorable outcomes for a score of 6:
\[ 6 + 1 + 1 = 8 \]

### Step 5: Total Favorable Outcomes
Now, we sum the favorable outcomes for both scores:
\[ 4 \text{ (for score 4)} + 8 \text{ (for score 6)} = 12 \]

### Step 6: Calculate Probability
The probability of getting a total score of 4 or 6 is given by:
\[ P(\text{score 4 or 6}) = \frac{\text{Total Favorable Outcomes}}{\text{Total Outcomes}} = \frac{12}{216} = \frac{1}{18} \]

### Final Answer
The probability of getting a total score of 4 or 6 when rolling the die three times is:
\[ \frac{1}{18} \]