Two dice are thrown simultaneously. Find the probability that a sum less than 7 will turn up on the upper faces.

Answer

To find the probability that the sum of the numbers on the upper faces of two thrown dice is less than 7, we can follow these steps:

### Step 1: Determine the Total Number of 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 Favorable Outcomes
Next, we need to find the combinations of the two dice that result in a sum less than 7. The possible sums and their combinations are:

- **Sum = 2**: (1, 1)
- **Sum = 3**: (1, 2), (2, 1)
- **Sum = 4**: (1, 3), (2, 2), (3, 1)
- **Sum = 5**: (1, 4), (2, 3), (3, 2), (4, 1)
- **Sum = 6**: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1)

Now, let's count these combinations:
- For sum = 2: 1 outcome
- For sum = 3: 2 outcomes
- For sum = 4: 3 outcomes
- For sum = 5: 4 outcomes
- For sum = 6: 5 outcomes

Adding these together gives us:
\[ 1 + 2 + 3 + 4 + 5 = 15 \text{ favorable outcomes} \]

### Step 3: Calculate the Probability
The probability \( P \) of an event is given by the formula:
\[ P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} \]
Substituting the values we found:
\[ P(\text{Sum < 7}) = \frac{15}{36} \]
We can simplify this fraction:
\[ P(\text{Sum < 7}) = \frac{5}{12} \]

### Final Answer
The probability that the sum of the numbers on the upper faces of the two dice is less than 7 is:
\[ \frac{5}{12} \]

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