Two dice are rolled. Find the probability that the sum of the numbers appears on the upper face of dice is equal to 9

Answer

To find the probability that the sum of the numbers appearing on the upper faces of two rolled dice is equal to 9, we can follow these steps:

### Step 1: Determine the total number of outcomes when rolling two dice.
When rolling two dice, each die has 6 faces. Therefore, the total number of possible outcomes when rolling two dice is calculated as:
\[ \text{Total outcomes} = 6 \times 6 = 36 \]

### Step 2: Identify the favorable outcomes that result in a sum of 9.
Next, we need to find all the combinations of the two dice that sum up to 9. The pairs that give a sum of 9 are:
- (3, 6)
- (4, 5)
- (5, 4)
- (6, 3)

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

### Step 3: Count the number of favorable outcomes.
From the pairs identified in Step 2, we can see that there are 4 favorable outcomes.

### Step 4: Calculate the probability.
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of outcomes. Thus, the probability \( P \) that the sum is 9 can be calculated as:
\[ P(\text{sum} = 9) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{36} \]

### Step 5: Simplify the probability.
Now, we simplify the fraction:
\[ P(\text{sum} = 9) = \frac{4}{36} = \frac{1}{9} \]

### Conclusion:
The probability that the sum of the numbers on the upper faces of the two dice is equal to 9 is \( \frac{1}{9} \).

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