Two dice are tossed. The probability that the total score is a prime number is :

Options

A. A) `1/6`
B. B) `5/12`
C. C) `1/2`
D. D) `7/9`

Answer

To find the probability that the total score when two dice are tossed is a prime number, we can follow these steps:

### Step 1: Determine the Sample Space
When two dice are tossed, each die has 6 faces. Therefore, the total number of possible outcomes (sample space) when tossing two dice is:
\[ N(S) = 6 \times 6 = 36 \]

### Step 2: Identify Prime Numbers
Next, we need to identify the prime numbers that can be obtained as the sum of the two dice. The possible sums when rolling two dice range from 2 (1+1) to 12 (6+6). The prime numbers in this range are:
- 2
- 3
- 5
- 7
- 11

### Step 3: List Outcomes for Each Prime Sum
Now, we will find all the combinations of dice rolls that result in these prime sums:

- **Sum = 2**: (1, 1) → 1 outcome
- **Sum = 3**: (1, 2), (2, 1) → 2 outcomes
- **Sum = 5**: (1, 4), (2, 3), (3, 2), (4, 1) → 4 outcomes
- **Sum = 7**: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) → 6 outcomes
- **Sum = 11**: (5, 6), (6, 5) → 2 outcomes

### Step 4: Count the Total Outcomes for Prime Sums
Now, we add the number of outcomes for each prime sum:
\[ 1 + 2 + 4 + 6 + 2 = 15 \]
So, the number of favorable outcomes (event E) is:
\[ N(E) = 15 \]

### Step 5: Calculate the Probability
The probability of the event E (total score is a prime number) is given by the formula:
\[ P(E) = \frac{N(E)}{N(S)} = \frac{15}{36} \]
This fraction can be simplified:
\[ P(E) = \frac{5}{12} \]

### Final Answer
Thus, the probability that the total score is a prime number when two dice are tossed is:
\[ \frac{5}{12} \]

---