Find the probability of getting the product of a perfect square (square of a natural number), when two dice are thrown together.

Answer

To find the probability of getting the product of a perfect square when two dice are thrown together, we can follow these steps:

### Step 1: Understand the Total Outcomes
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing two dice is:
\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

### Step 2: Identify Perfect Squares
The perfect squares that can be formed from the products of the numbers on the dice are:
- \(1^2 = 1\)
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(4^2 = 16\)
- \(5^2 = 25\)
- \(6^2 = 36\)

### Step 3: List Possible Combinations
Now we need to find the pairs of dice rolls that yield these perfect squares:

1. **For \(1\)**:
- \( (1, 1) \) → Product = 1 (Perfect square)

2. **For \(4\)**:
- \( (1, 4) \) → Product = 4 (Perfect square)
- \( (2, 2) \) → Product = 4 (Perfect square)
- \( (4, 1) \) → Product = 4 (Perfect square)

3. **For \(9\)**:
- \( (3, 3) \) → Product = 9 (Perfect square)

4. **For \(16\)**:
- \( (4, 4) \) → Product = 16 (Perfect square)

5. **For \(25\)**:
- \( (5, 5) \) → Product = 25 (Perfect square)

6. **For \(36\)**:
- \( (6, 6) \) → Product = 36 (Perfect square)

### Step 4: Count the Successful Outcomes
Now, we count the successful outcomes that yield a perfect square:
- From \(1\): 1 outcome
- From \(4\): 3 outcomes
- From \(9\): 1 outcome
- From \(16\): 1 outcome
- From \(25\): 1 outcome
- From \(36\): 1 outcome

Total successful outcomes = \(1 + 3 + 1 + 1 + 1 + 1 = 8\)

### Step 5: Calculate the Probability
The probability \(P\) of getting a product that is a perfect square is given by the ratio of successful outcomes to total outcomes:
\[ P = \frac{\text{Number of Successful Outcomes}}{\text{Total Outcomes}} = \frac{8}{36} \]

### Step 6: Simplify the Probability
Now, we simplify the fraction:
\[ P = \frac{8}{36} = \frac{2}{9} \]

### Final Answer
Thus, the probability of getting the product of a perfect square when two dice are thrown is:
\[ \boxed{\frac{2}{9}} \]