In a simultaneous throw of a pair of dice, find the probability of getting:
a sum less than 8.

Answer

To find the probability of getting a sum less than 8 when throwing a pair of dice, 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 the favorable outcomes for sums less than 8
Now, we will list all the combinations of the two dice that give a sum less than 8. The possible sums and their combinations are as follows:

- **Sum = 2**: (1, 1) → 1 way
- **Sum = 3**: (1, 2), (2, 1) → 2 ways
- **Sum = 4**: (1, 3), (2, 2), (3, 1) → 3 ways
- **Sum = 5**: (1, 4), (2, 3), (3, 2), (4, 1) → 4 ways
- **Sum = 6**: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) → 5 ways
- **Sum = 7**: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) → 6 ways

Now, we can add these favorable outcomes:
\[ 1 + 2 + 3 + 4 + 5 + 6 = 21 \]

### Step 3: Calculate the probability
The probability of getting a sum less than 8 is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{21}{36} \]

### Step 4: Simplify the probability
To simplify \(\frac{21}{36}\), we can divide both the numerator and the denominator by their greatest common divisor, which is 3:
\[ \frac{21 \div 3}{36 \div 3} = \frac{7}{12} \]

### Final Answer
Thus, the probability of getting a sum less than 8 when throwing a pair of dice is:
\[ \frac{7}{12} \]

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