Two different dice one thrown together. Find the probability that the numbers obtained :
have a sum less than 7

Answer

To solve the problem of finding the probability that the sum of the numbers obtained from throwing two different dice is less than 7, we can follow these steps:

### Step-by-Step Solution:

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

2. **Identify Favorable Outcomes**:
We need to find the pairs of numbers (from the two dice) that sum to less than 7. Let's list them:
- For the first die showing 1:
- (1, 1) → Sum = 2
- (1, 2) → Sum = 3
- (1, 3) → Sum = 4
- (1, 4) → Sum = 5
- (1, 5) → Sum = 6
- For the first die showing 2:
- (2, 1) → Sum = 3
- (2, 2) → Sum = 4
- (2, 3) → Sum = 5
- (2, 4) → Sum = 6
- For the first die showing 3:
- (3, 1) → Sum = 4
- (3, 2) → Sum = 5
- (3, 3) → Sum = 6
- For the first die showing 4:
- (4, 1) → Sum = 5
- (4, 2) → Sum = 6
- For the first die showing 5:
- (5, 1) → Sum = 6

The pairs that yield sums less than 7 are:
- (1, 1), (1, 2), (1, 3), (1, 4), (1, 5)
- (2, 1), (2, 2), (2, 3), (2, 4)
- (3, 1), (3, 2)
- (4, 1)
- (5, 1)

Counting these pairs, we have:
- 5 outcomes from the first die showing 1
- 4 outcomes from the first die showing 2
- 3 outcomes from the first die showing 3
- 2 outcomes from the first die showing 4
- 1 outcome from the first die showing 5

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

3. **Calculate the Probability**:
The probability \( P \) of getting a sum less than 7 is given by the formula:
\[ P(\text{Sum} < 7) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{15}{36} \]

4. **Simplify the Probability**:
We can simplify \( \frac{15}{36} \) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3:
\[ P(\text{Sum} < 7) = \frac{15 \div 3}{36 \div 3} = \frac{5}{12} \]

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