Math Problem Statement
(ii)When two dice are thrown, A be the event of having sum of the numbers in the dice to be 8, B be the event of having product of numbers in the dice to be 12. Are A,B mutually exclusive?
Solution
To determine whether the events and are mutually exclusive, we need to verify if they can occur simultaneously.
Definitions:
- Event : The sum of the numbers on two dice is 8.
- Event : The product of the numbers on two dice is 12.
Step 1: Possible Outcomes for Event
For the sum of two dice to be 8: The pairs are:
Step 2: Possible Outcomes for Event
For the product of two dice to be 12: The pairs are:
Step 3: Intersection of and
The events and are mutually exclusive if they have no outcomes in common.
From :
From :
The common outcomes (intersection) are:
Conclusion:
Since , the events and are not mutually exclusive because they can occur simultaneously (e.g., and ).
Would you like further clarification or a detailed explanation of any step?
Here are 5 related questions to expand your understanding:
- How do we calculate probabilities of events for two dice?
- Can you explain what it means for events to be independent rather than mutually exclusive?
- How would the answer change if the dice were weighted or biased?
- What is the probability of each individual event, and ?
- How would this problem change if we used three dice instead of two?
Tip: Always write down all possible outcomes for small sample spaces like dice rolls to ensure you don’t miss any cases.
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Math Problem Analysis
Mathematical Concepts
Probability
Set Theory
Dice Probabilities
Formulas
Sum of numbers on dice
Product of numbers on dice
Theorems
Mutually Exclusive Events
Suitable Grade Level
Grades 9-12
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