Math Problem Statement
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.
Solution
To solve the problem, let's break it down step by step:
Step 1: Identify the possible outcomes.
You know that the sum of the results from rolling the two dice is .
The possible outcomes for two dice rolls that sum to 12 are:
This is the only pair where the sum of the two numbers is 12, since the largest number possible on a die is 6.
Step 2: Analyze the possible values for .
Given that the sum is , the only outcome is , meaning and .
Step 3: Check if is odd.
The value of is 6, which is even, not odd.
Step 4: Calculate the probability.
Since is the only possible value and it's even, the probability that is odd is:
Thus, the probability that is odd given that is 0.
Would you like more details, or have any questions about the solution?
Here are five related questions to expand on this:
- What is the probability that ?
- What is the probability that both dice show an odd number?
- How does the probability change if you know is greater than ?
- What is the probability that the sum of two rolls is at least 10?
- How do you calculate conditional probabilities when rolling dice?
Tip: When rolling dice, identifying all possible outcomes before applying conditions is key to correctly calculating probabilities.
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Math Problem Analysis
Mathematical Concepts
Probability
Conditional Probability
Combinatorics
Formulas
P(A|B) = P(A and B) / P(B)
Theorems
Basic Probability Theorem
Suitable Grade Level
Grades 6-8