To find the probability of rolling two dice such that the sum of the numbers is an odd number, we can proceed as follows:

Total possible outcomes:

• When rolling two dice, each die has 6 faces, so the total number of outcomes is: $6 \times 6 = 36$

Favorable outcomes (sum is odd):

• A sum of two numbers is odd when one number is odd and the other is even. This can happen in two ways:
  1. The first die shows an odd number, and the second die shows an even number.
  2. The first die shows an even number, and the second die shows an odd number.

Odd numbers on a die: 1, 3, 5 (3 options)

Even numbers on a die: 2, 4, 6 (3 options)

• In case 1 (odd on the first die, even on the second), there are $3 \times 3 = 9$ possible outcomes.
• In case 2 (even on the first die, odd on the second), there are also $3 \times 3 = 9$ possible outcomes.

So, the total number of favorable outcomes is: $9 + 9 = 18$

Probability:

The probability is the ratio of favorable outcomes to total outcomes: $\frac{18}{36} = \frac{1}{2}$

Thus, the probability that the sum of the numbers is odd is: $\boxed{\frac{1}{2}}$

Would you like more details or have any questions?

Here are five related questions to expand your understanding:

1. What is the probability that the sum of the numbers is even?
2. What is the probability that the sum of the numbers is greater than 9?
3. What is the probability of rolling a double (both dice showing the same number)?
4. What is the probability of rolling a sum of 7?
5. What is the probability of rolling two odd numbers?

Tip: When calculating probabilities with dice, always start by counting the total number of possible outcomes, which is typically the product of the number of faces on each die.