I roll a fair die twice and obtain two numbers 

X
1
X 
1
​
 = result of the first roll and 
X
2
X 
2
​
  = result of the second roll. 

Given that I know 
X
1
+
X
2
=
12
X 
1
​
 +X 
2
​
 =12,   what is the probability that 
X
1
X 
1
​
  is odd.


Given the problem:

- \( X_1 \) is the result of the first die roll.
- \( X_2 \) is the result of the second die roll.
- You know that \( X_1 + X_2 = 12 \).
- You want to find the probability that \( X_1 \) is odd.

### Step 1: List Possible Combinations of \( X_1 \) and \( X_2 \)

Since \( X_1 + X_2 = 12 \), the possible pairs of \( (X_1, X_2) \), where each number comes from a fair die (numbers 1 through 6), are:

\[
X_1 + X_2 = 12 \implies (X_1, X_2) = (6, 6)
\]

The only combination that satisfies this equation is \( X_1 = 6 \) and \( X_2 = 6 \).

### Step 2: Determine if \( X_1 \) is Odd

For \( (X_1, X_2) = (6, 6) \), \( X_1 = 6 \), which is **even**, not odd.

### Step 3: Conclusion

Since there is only one possible outcome, \( (X_1, X_2) = (6, 6) \), and \( X_1 \) is even, the probability that \( X_1 \) is odd is:

\[
P(X_1 \text{ is odd}) = 0
\]

The probability is **0**.

---

Would you like further clarification or have any additional questions? Here are some related questions you might consider:

1. What is the probability that \( X_2 \) is odd given the same conditions?
2. How does the result change if I roll a die three times and sum the results?
3. What is the probability of obtaining an even number for both dice rolls?
4. How would the problem change if the sum were a different number?
5. What is the probability that both dice rolls give the same result?

**Tip:** When working with probabilities involving dice, listing all possible outcomes helps identify patterns and constraints quickly.

I roll a fair die twice and obtain two numbers X1 = result of the first roll and X2 = result of the second roll. Given that I know X1 + X2 = 12, what is the probability that X1 is odd?

Learn how to find the probability that the result of the first die roll is odd when X1 + X2 = 12. This probability problem involves basic combinatorics and probability theory, suitable for students in Grades 6-8.

Math Problem Statement

Given that I know X 1 + X 2

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