Institution: Bangladesh Bank Exam: Officer (General) |-Sonali | Janata | Bangladesh Krishi_Bangladesh Development bank ২৪.০৫.২০১৯ ||
Topic: Arithmetic Mathematics (পাটিগণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

To find the probability of rolling a total of at least 6 with three dice, we first need to calculate the total possible outcomes when three dice are thrown. Each die has 6 faces, so for three dice, the total number of outcomes is \(6^3 = 216\).

Next, we'll find the outcomes where the total is less than 6. The minimum sum achievable with three dice is 3 (when all dice show 1). The possible sums less than 6 are 3, 4, and 5:

1. **Sum = 3:** Only one outcome: (1, 1, 1).
2. **Sum = 4:** The combinations are (1, 1, 2) in all permutations, which gives us 3 different arrangements: (1, 1, 2), (1, 2, 1), (2, 1, 1). Thus, there are 3 outcomes.
3. **Sum = 5:** The combinations are (1, 1, 3), (1, 2, 2) in all permutations. The arrangements for (1, 1, 3) give 3 outcomes. The arrangements for (1, 2, 2) give 3 outcomes as well. Altogether, we have \(3 + 3 = 6\) outcomes.

Adding these, the total outcomes with sums less than 6 is \(1 + 3 + 6 = 10\).

Now, we subtract this from the total outcomes: \(216 - 10 = 206\). Therefore, the probability of getting a total of at least 6 is given by:

\[
P(\text{at least 6}) = \frac{206}{216} = \frac{103}{108}.
\]

Thus, the correct answer is option A: \( \frac{103}{108} \).