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To solve the problem of finding the probability that, when rolling a standard die four times, an odd number only shows up twice, we need to use the concept of binomial probability.

### Breakdown of the Problem:

1. **Possible Outcomes for Each Roll:**
   - The standard die has 6 sides: {1, 2, 3, 4, 5, 6}.
   - Odd numbers: {1, 3, 5} (there are 3 odd numbers).
   - Even numbers: {2, 4, 6} (there are 3 even numbers).
   
   So, the probability of rolling an odd number on any single roll is:
   \[
   P(\text{Odd}) = \frac{3}{6} = \frac{1}{2}
   \]
   Similarly, the probability of rolling an even number is also \( \frac{1}{2} \).

2. **Using Binomial Probability Formula:**
   We are interested in the event where exactly two of the four rolls are odd numbers.

   The binomial probability formula is:
   \[
   P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
   \]
   Where:
   - \( n \) is the number of trials (4 rolls),
   - \( k \) is the number of successes (2 odd numbers),
   - \( p \) is the probability of success on a single trial (\( \frac{1}{2} \)).

3. **Substitute Values:**
   Here, \( n = 4 \), \( k = 2 \), and \( p = \frac{1}{2} \). The formula becomes:
   \[
   P(\text{exactly 2 odds}) = \binom{4}{2} \left( \frac{1}{2} \right)^2 \left( \frac{1}{2} \right)^{4-2}
   \]
   Simplifying:
   \[
   P(\text{exactly 2 odds}) = \binom{4}{2} \left( \frac{1}{2} \right)^4
   \]

Let me calculate this step by step.The probability that when rolling a standard die four times, an odd number shows up exactly twice is \( 0.375 \), or 37.5%.

Would you like more details on this calculation or clarification on any steps?

Here are some related questions you might find interesting:
1. What is the probability of getting exactly three odd numbers in four rolls of a die?
2. How would the probability change if the number of rolls increased to six?
3. What is the expected number of odd numbers when rolling a die four times?
4. How does the binomial probability formula change for events with unequal success probabilities?
5. How would you calculate the probability of getting at least two odd numbers in four rolls?

**Tip:** When solving probability problems involving multiple independent events, binomial distribution is often a powerful and efficient tool.

Find the probability that when rolling a standard die 4 times, an odd number only shows up twice.

Learn how to find the probability of rolling an odd number exactly twice in 4 rolls of a standard die. This solution uses the binomial distribution formula and involves key concepts in probability and combinatorics. Suitable for Grades 10-12.

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