In a single throw of two dice , what is the probability of
a total less than 10

Answer

To find the probability of getting a total less than 10 when throwing two dice, we can follow these steps:

### Step 1: Determine the Sample Space
When two dice are thrown, each die has 6 faces. Therefore, the total number of possible outcomes (sample space) when throwing two dice is:
\[ 6 \times 6 = 36 \]

### Step 2: Identify the Favorable Outcomes
Next, we need to find the outcomes where the sum of the two dice is less than 10. We can list the possible sums:

- **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)
- **Sum = 7**: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
- **Sum = 8**: (2,6), (3,5), (4,4), (5,3), (6,2)
- **Sum = 9**: (3,6), (4,5), (5,4), (6,3)

Now, we can count the outcomes for sums less than 10:
- 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
- For sum = 7: 6 outcomes
- For sum = 8: 5 outcomes
- For sum = 9: 4 outcomes

Adding these, we get:
\[ 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 = 30 \text{ favorable outcomes} \]

### Step 3: Calculate the Probability
The probability of an event is given by the formula:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
Substituting the values we found:
\[ P(\text{sum < 10}) = \frac{30}{36} \]

### Step 4: Simplify the Probability
We can simplify \(\frac{30}{36}\):
\[ P(\text{sum < 10}) = \frac{5}{6} \]

### Final Answer
Thus, the probability of getting a total less than 10 when throwing two dice is:
\[ \frac{5}{6} \]

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