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 AA and BB are mutually exclusive, we need to verify if they can occur simultaneously.

Definitions:

  • Event AA: The sum of the numbers on two dice is 8.
  • Event BB: The product of the numbers on two dice is 12.

Step 1: Possible Outcomes for Event AA

For the sum of two dice to be 8: (x,y) where x+y=8,  x,y{1,2,3,4,5,6}.(x, y) \text{ where } x + y = 8, \; x, y \in \{1, 2, 3, 4, 5, 6\}. The pairs are: A={(2,6),(3,5),(4,4),(5,3),(6,2)}.A = \{(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)\}.

Step 2: Possible Outcomes for Event BB

For the product of two dice to be 12: (x,y) where xy=12,  x,y{1,2,3,4,5,6}.(x, y) \text{ where } x \cdot y = 12, \; x, y \in \{1, 2, 3, 4, 5, 6\}. The pairs are: B={(2,6),(3,4),(4,3),(6,2)}.B = \{(2, 6), (3, 4), (4, 3), (6, 2)\}.

Step 3: Intersection of AA and BB

The events AA and BB are mutually exclusive if they have no outcomes in common.

From AA: A={(2,6),(3,5),(4,4),(5,3),(6,2)}.A = \{(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)\}.

From BB: B={(2,6),(3,4),(4,3),(6,2)}.B = \{(2, 6), (3, 4), (4, 3), (6, 2)\}.

The common outcomes (intersection) are: AB={(2,6),(6,2)}.A \cap B = \{(2, 6), (6, 2)\}.

Conclusion:

Since ABA \cap B \neq \emptyset, the events AA and BB are not mutually exclusive because they can occur simultaneously (e.g., (2,6)(2, 6) and (6,2)(6, 2)).


Would you like further clarification or a detailed explanation of any step?
Here are 5 related questions to expand your understanding:

  1. How do we calculate probabilities of events for two dice?
  2. Can you explain what it means for events to be independent rather than mutually exclusive?
  3. How would the answer change if the dice were weighted or biased?
  4. What is the probability of each individual event, AA and BB?
  5. 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