Two dice are thrown simultaneously. Find the probability of getting an odd number as the sum.

Answer

To find the probability of getting an odd number as the sum when two dice are thrown simultaneously, we can follow these steps:

### Step 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:
\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

### Step 2: Identify the Conditions for an Odd Sum
The sum of two numbers is odd if one number is odd and the other number is even. Therefore, we need to consider the combinations of odd and even numbers on the dice.

### Step 3: List the Odd and Even Numbers on a Die
The numbers on a die are:
- Odd numbers: 1, 3, 5
- Even numbers: 2, 4, 6

### Step 4: Count the Combinations for Odd Sum
We can have the following combinations:
1. **First die is odd, second die is even:**
- Odd choices for the first die: 1, 3, 5 (3 choices)
- Even choices for the second die: 2, 4, 6 (3 choices)
- Total combinations = \(3 \times 3 = 9\)

2. **First die is even, second die is odd:**
- Even choices for the first die: 2, 4, 6 (3 choices)
- Odd choices for the second die: 1, 3, 5 (3 choices)
- Total combinations = \(3 \times 3 = 9\)

### Step 5: Calculate the Total Combinations for Odd Sums
Now, we add the combinations from both cases:
\[ \text{Total combinations for odd sums} = 9 + 9 = 18 \]

### Step 6: Calculate the Probability
The probability of getting an odd sum is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{18}{36} = \frac{1}{2} \]

### Final Answer
The probability of getting an odd number as the sum when two dice are thrown is:
\[ \frac{1}{2} \]

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