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 X1+X2=12X_1 + X_2 = 12.

The possible outcomes for two dice rolls that sum to 12 are:

  • (6,6)(6, 6)

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 X1X_1.

Given that the sum is X1+X2=12X_1 + X_2 = 12, the only outcome is (6,6)(6, 6), meaning X1=6X_1 = 6 and X2=6X_2 = 6.

Step 3: Check if X1X_1 is odd.

The value of X1X_1 is 6, which is even, not odd.

Step 4: Calculate the probability.

Since X1=6X_1 = 6 is the only possible value and it's even, the probability that X1X_1 is odd is:

P(X1 is odd)=0P(X_1 \text{ is odd}) = 0

Thus, the probability that X1X_1 is odd given that X1+X2=12X_1 + X_2 = 12 is 0.


Would you like more details, or have any questions about the solution?

Here are five related questions to expand on this:

  1. What is the probability that X1+X2=7X_1 + X_2 = 7?
  2. What is the probability that both dice show an odd number?
  3. How does the probability change if you know X1X_1 is greater than X2X_2?
  4. What is the probability that the sum of two rolls is at least 10?
  5. 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