To solve this, we'll go through a step-by-step process:

Step 1: Understand the composition of the dice

Each die has six sides:

• Odd numbers: $\{1, 3, 5\}$ (3 outcomes)
• Even numbers: $\{2, 4, 6\}$ (3 outcomes)

Step 2: Define the problem

We need the probability of rolling one odd number and two even numbers with three dice.

Step 3: Calculate the number of favorable outcomes

For one odd number and two even numbers, we:

1. Choose one die to show an odd number (there are 3 odd outcomes).
2. Choose two dice to show even numbers (there are 3 even outcomes for each die).

Thus, for each combination:

• Odd outcome: $3$ possibilities
• Even outcomes: $3 \times 3 = 9$ possibilities

The total number of favorable outcomes is: $3 \times 9 = 27$ We also need to account for the different ways the dice could land, so we multiply by the number of ways to choose which die will be odd and which two will be even. This is a combination of three dice taken one at a time, which gives: $\text{Number of ways to choose 1 odd die out of 3 dice} = \binom{3}{1} = 3$

So, the total number of favorable outcomes is: $27 \times 3 = 81$

Step 4: Calculate the total possible outcomes

The total number of outcomes for rolling three dice is: $6 \times 6 \times 6 = 216$

Step 5: Calculate the probability

The probability is the ratio of favorable outcomes to the total outcomes: $P(\text{1 odd, 2 even}) = \frac{81}{216} = \frac{3}{8}$

Thus, the probability of getting one odd number and two even numbers when rolling three dice is $\frac{3}{8}$.

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Would you like further details on this calculation or have any questions?

Here are five related questions:

1. What is the probability of getting two odd numbers and one even number?
2. How does the probability change if we use four dice instead of three?
3. What is the probability of getting all three odd numbers?
4. Can we calculate the expected number of odd numbers if we roll the dice multiple times?
5. What is the probability of getting a sum of 10 with three dice?

Tip: When solving probability problems involving dice, it helps to break down the situation into individual events and consider both favorable outcomes and total outcomes.