Two fair dice are tossed simultaneously. What is the probability that the product of the numbers that land showing is at at least 25 ?

Options

A. `(1)/(36)`
B. `(1)/(12)`
C. `(1)/(9)`
D. `(1)/(4)`

Explanation

To find the probability that the product of the numbers shown on two tossed fair dice is at least 25, we can follow these steps:

### Step 1: Determine the total number of outcomes
When two dice are tossed, each die has 6 faces. Therefore, the total number of outcomes when two dice are rolled is:
\[ 6 \times 6 = 36 \]

### Step 2: Identify the favorable outcomes
Next, we need to find the combinations of the two dice that yield a product of at least 25. We can list the possible products for each combination of dice rolls:

- For (1,1) to (1,6): Products are 1, 2, 3, 4, 5, 6
- For (2,1) to (2,6): Products are 2, 4, 6, 8, 10, 12
- For (3,1) to (3,6): Products are 3, 6, 9, 12, 15, 18
- For (4,1) to (4,6): Products are 4, 8, 12, 16, 20, 24
- For (5,1) to (5,6): Products are 5, 10, 15, 20, 25, 30
- For (6,1) to (6,6): Products are 6, 12, 18, 24, 30, 36

Now, we need to identify which of these products are at least 25:
- From (5,5): Product = 25
- From (5,6): Product = 30
- From (6,5): Product = 30
- From (6,6): Product = 36

Thus, the favorable outcomes are:
1. (5,5)
2. (5,6)
3. (6,5)
4. (6,6)

This gives us a total of 4 favorable outcomes.

### Step 3: Calculate the probability
The probability \( P \) of an event is given by the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
Substituting the values we found:
\[ P = \frac{4}{36} = \frac{1}{9} \]

### Conclusion
The probability that the product of the numbers shown on the two dice is at least 25 is:
\[ \frac{1}{9} \]