Math Problem Statement

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.

Solution

Given the problem:

  • X1X_1 is the result of the first die roll.
  • X2X_2 is the result of the second die roll.
  • You know that X1+X2=12X_1 + X_2 = 12.
  • You want to find the probability that X1X_1 is odd.

Step 1: List Possible Combinations of X1X_1 and X2X_2

Since X1+X2=12X_1 + X_2 = 12, the possible pairs of (X1,X2)(X_1, X_2), where each number comes from a fair die (numbers 1 through 6), are:

X1+X2=12    (X1,X2)=(6,6)X_1 + X_2 = 12 \implies (X_1, X_2) = (6, 6)

The only combination that satisfies this equation is X1=6X_1 = 6 and X2=6X_2 = 6.

Step 2: Determine if X1X_1 is Odd

For (X1,X2)=(6,6)(X_1, X_2) = (6, 6), X1=6X_1 = 6, which is even, not odd.

Step 3: Conclusion

Since there is only one possible outcome, (X1,X2)=(6,6)(X_1, X_2) = (6, 6), and X1X_1 is even, the probability that X1X_1 is odd is:

P(X1 is odd)=0P(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 X2X_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.

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Math Problem Analysis

Mathematical Concepts

Probability
Combinatorics

Formulas

P(event) = favorable outcomes / total outcomes

Theorems

Basic Probability Theorem

Suitable Grade Level

Grades 6-8