Two persons each make a single throw with a- pair of dice. The probability that the throws are equal, is

Options

A. `(73)/(648)`
B. `(73)/(1296)`
C. `(182)/(648)`
D. none of these

Explanation

To find the probability that two persons, each throwing a pair of dice, get equal outcomes, we can follow these steps:

### Step 1: Determine the total outcomes for each person
When a person throws a pair of dice, there are 6 faces on each die. Therefore, the total number of outcomes when throwing a pair of dice is:
\[ 6 \times 6 = 36 \]
Thus, both persons (let's call them Person A and Person B) have 36 possible outcomes each.

### Step 2: Calculate the total outcomes for both persons
Since both persons throw their dice independently, the total number of combined outcomes is:
\[ 36 \times 36 = 1296 \]

### Step 3: Identify the favorable outcomes for equal throws
Next, we need to find the number of ways in which both persons can get equal outcomes. The possible sums when throwing two dice range from 2 to 12. We will count the number of ways to achieve each sum:

- **Sum = 2:** (1,1) → 1 way
- **Sum = 3:** (1,2), (2,1) → 2 ways
- **Sum = 4:** (1,3), (2,2), (3,1) → 3 ways
- **Sum = 5:** (1,4), (2,3), (3,2), (4,1) → 4 ways
- **Sum = 6:** (1,5), (2,4), (3,3), (4,2), (5,1) → 5 ways
- **Sum = 7:** (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 ways
- **Sum = 8:** (2,6), (3,5), (4,4), (5,3), (6,2) → 5 ways
- **Sum = 9:** (3,6), (4,5), (5,4), (6,3) → 4 ways
- **Sum = 10:** (4,6), (5,5), (6,4) → 3 ways
- **Sum = 11:** (5,6), (6,5) → 2 ways
- **Sum = 12:** (6,6) → 1 way

### Step 4: Calculate the total number of favorable outcomes
Now, we sum all the favorable outcomes:
\[ 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36 \]

### Step 5: Calculate the probability
The probability that both throws are equal is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{equal throws}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{36}{1296} \]

### Step 6: Simplify the probability
To simplify:
\[ P(\text{equal throws}) = \frac{36 \div 36}{1296 \div 36} = \frac{1}{36} \]

### Final Answer
Thus, the probability that the throws are equal is:
\[ \frac{1}{36} \]

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