Question

Two dice are thrown. Defined are the following two events A and B: \[ A = \{(x, y) : x + y = 9\}, \quad B = \{(x, y) : x \neq 3\}, \] where \( (x, y) \) denote a point in the sample space. Check if events \( A \) and \( B \) are independent or mutually exclusive.

Hint:

Events \( A \) and \( B \) are independent if \( P(A \cap B) = P(A)P(B) \) and mutually exclusive if \( P(A \cap B) = 0 \).

Answer 1

This problem involves two events derived from the outcomes of rolling two dice. The sample space encompasses all ordered pairs \( (x, y) \), where \( x \) and \( y \) can each be any integer from 1 to 6.

1. Event Definitions:
- Event \( A \) is defined as the set of outcomes where the sum of the two dice is 9: \( A = \{(x, y) : x + y = 9\} \).
- Event \( B \) is defined as the set of outcomes where the first die's result is not 3: \( B = \{(x, y) : x eq 3\} \).

2. Total Possible Outcomes:
With two dice, each having 6 faces, the total number of distinct outcomes is \( 6 \times 6 = 36 \).

3. Outcomes Constituting Event A:
The pairs summing to 9 are: \( A = \{(3, 6), (4, 5), (5, 4), (6, 3)\} \). Thus, the number of outcomes in \( A \) is \( n(A) = 4 \).

4. Outcomes Constituting Event B:
Event \( B \) includes all outcomes except those where the first die is 3. There are 6 outcomes where \( x = 3 \) (i.e., (3,1) through (3,6)). Therefore, the number of outcomes in \( B \) is \( 36 - 6 = 30 \). Thus, \( n(B) = 30 \).

5. Intersection of Events A and B ( \( A \cap B \) ):
We seek outcomes present in both \( A \) and \( B \). From event \( A \), the only outcome where \( x = 3 \) is \( (3,6) \). Removing this from \( A \) gives the intersection: \( A \cap B = A \setminus \{(3,6)\} = \{(4,5), (5,4), (6,3)\} \). The number of outcomes in the intersection is \( n(A \cap B) = 3 \).

6. Mutual Exclusivity Check:
Events are mutually exclusive if their intersection is empty (\( A \cap B = \emptyset \)). Since \( A \cap B \) contains 3 outcomes, the events are not mutually exclusive.

7. Independence Check:
Events \( A \) and \( B \) are independent if the probability of their intersection equals the product of their individual probabilities: \( P(A \cap B) = P(A) \cdot P(B) \).
The probabilities are calculated as follows:
- \( P(A) = \frac{n(A)}{\text{Total Outcomes}} = \frac{4}{36} = \frac{1}{9} \)
- \( P(B) = \frac{n(B)}{\text{Total Outcomes}} = \frac{30}{36} = \frac{5}{6} \)
- \( P(A \cap B) = \frac{n(A \cap B)}{\text{Total Outcomes}} = \frac{3}{36} = \frac{1}{12} \)
Now, we compare: \( P(A) \cdot P(B) = \frac{1}{9} \cdot \frac{5}{6} = \frac{5}{54} \).
Since \( \frac{1}{12} eq \frac{5}{54} \) (which is \( \frac{4.5}{54} eq \frac{5}{54} \)), the events are not independent.

8. Final Determination:
Based on the checks, events \( A \) and \( B \) are neither independent nor mutually exclusive.

Final Answer:
The events \( A \) and \( B \) are neither mutually exclusive nor independent.