Math Problem Statement
Two dice are rolled. Determine the probability of the following. ("Doubles" means both dice show the same number.) Rolling an even number or doubles
Solution
To solve this, we calculate the probability of rolling an even number or doubles using the principles of probability. Here's the step-by-step explanation:
Total Outcomes
When two dice are rolled, the total number of outcomes is:
Event 1: Rolling an Even Number
An even number can appear on either die. The even numbers are . For each die, there are even outcomes. The number of outcomes where at least one die shows an even number is calculated using complementary counting (subtracting cases where both dice show odd numbers).
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Both dice odd: Numbers , which means choices for each die:
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At least one die even:
So, there are outcomes where at least one die shows an even number.
Event 2: Rolling Doubles
Doubles occur when both dice show the same number. The possible outcomes for doubles are: This gives outcomes.
Event 3: Intersection of Event 1 and Event 2
The intersection of "even number" and "doubles" occurs when the doubles are also even numbers. These outcomes are: This gives outcomes.
Applying the Inclusion-Exclusion Principle
The probability of "even number or doubles" is calculated as:
Substitute the values:
Final Answer:
The probability of rolling an even number or doubles is:
Let me know if you would like a detailed breakdown or if you have questions!
Related Questions:
- What is the probability of rolling only doubles?
- What is the probability of rolling two even numbers?
- What is the probability of rolling an odd number on both dice?
- What is the probability of rolling a sum greater than 8?
- How does the probability change if one die is weighted?
Tip:
When calculating probabilities of combined events, always consider whether they overlap to avoid double-counting!
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Math Problem Analysis
Mathematical Concepts
Probability
Inclusion-Exclusion Principle
Combinatorics
Formulas
Total outcomes = 6 × 6 = 36
P(A or B) = P(A) + P(B) - P(A and B)
Theorems
Inclusion-Exclusion Principle
Suitable Grade Level
Grades 8-10