Two dice are thrown at the same time. Write down all the possible outcomes. Find the probability of getting the sum of two numbers appearing on the top of the dice as:
less than 10

Answer

To solve the problem of finding the probability of getting a sum of two numbers appearing on the top of two thrown dice that is less than 10, we will follow these steps:

### Step 1: List all possible outcomes when two dice are thrown.
Each die has 6 faces numbered from 1 to 6. When two dice are thrown, the total number of outcomes can be calculated as follows:

- The first die can show any of the 6 numbers.
- The second die can also show any of the 6 numbers.

Thus, the total number of outcomes is:
\[ 6 \times 6 = 36 \]

The possible outcomes can be represented as ordered pairs (a, b), where 'a' is the result from the first die and 'b' is the result from the second die. The complete list of outcomes is:
- (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)
- (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)
- (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)
- (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)
- (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
- (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)

### Step 2: Identify the outcomes where the sum is less than 10.
Next, we need to find all the pairs (a, b) where the sum \( a + b < 10 \).

- For (1, _): The sums are 2, 3, 4, 5, 6, 7 (6 outcomes)
- For (2, _): The sums are 3, 4, 5, 6, 7, 8 (6 outcomes)
- For (3, _): The sums are 4, 5, 6, 7, 8, 9 (6 outcomes)
- For (4, _): The sums are 5, 6, 7, 8, 9 (5 outcomes)
- For (5, _): The sums are 6, 7, 8, 9 (4 outcomes)
- For (6, _): The sums are 7, 8, 9 (3 outcomes)

Now, we can count the total number of outcomes where the sum is less than 10:
- From (1, _): 6 outcomes
- From (2, _): 6 outcomes
- From (3, _): 6 outcomes
- From (4, _): 5 outcomes
- From (5, _): 4 outcomes
- From (6, _): 3 outcomes

Adding these gives:
\[ 6 + 6 + 6 + 5 + 4 + 3 = 30 \]

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

### Step 4: Simplify the probability.
To simplify \( \frac{30}{36} \):
\[ \frac{30 \div 6}{36 \div 6} = \frac{5}{6} \]

### Final Answer:
The probability of getting the sum of two numbers appearing on the top of the dice as less than 10 is:
\[ \frac{5}{6} \]