Three dice are thrown simultaneously. The probability of getting a total of atleast 6 is

Options

A. `5/108`
B. `103/108`
C. `7/108`
D. `53/54`

Answer

To find the probability of getting a total of at least 6 when three dice are thrown simultaneously, we can follow these steps:

### Step 1: Determine the total number of outcomes
When throwing three dice, each die has 6 faces. Therefore, the total number of outcomes when throwing three dice is calculated as follows:

\[ \text{Total outcomes} = 6 \times 6 \times 6 = 216 \]

### Step 2: Find the number of outcomes with a sum less than 6
Next, we need to find the number of outcomes where the sum of the three dice is less than 6. The possible combinations that yield a sum less than 6 are:

1. (1, 1, 1) → Sum = 3
2. (1, 1, 2) → Sum = 4
3. (1, 2, 1) → Sum = 4
4. (2, 1, 1) → Sum = 4
5. (1, 3, 1) → Sum = 5
6. (1, 1, 3) → Sum = 5
7. (2, 2, 1) → Sum = 5
8. (2, 1, 2) → Sum = 5
9. (1, 2, 2) → Sum = 5
10. (3, 1, 1) → Sum = 5

Counting these combinations, we find that there are a total of 10 outcomes where the sum is less than 6.

### Step 3: Calculate the number of outcomes with a sum of at least 6
To find the number of outcomes where the sum is at least 6, we subtract the number of outcomes with a sum less than 6 from the total number of outcomes:

\[ \text{Outcomes with sum at least 6} = \text{Total outcomes} - \text{Outcomes with sum less than 6} \]
\[ = 216 - 10 = 206 \]

### Step 4: Calculate the probability
The probability of getting a sum of at least 6 is given by the ratio of the number of favorable outcomes to the total number of outcomes:

\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{206}{216} \]

### Step 5: Simplify the probability
To simplify \(\frac{206}{216}\):

\[ \frac{206}{216} = \frac{103}{108} \]

Thus, the probability of getting a total of at least 6 when three dice are thrown is:

\[ \frac{103}{108} \]

### Final Answer
The probability of getting a total of at least 6 is \(\frac{103}{108}\).