Question

Two dice are thrown and the sum of the numbers which come up on the dice is noted. Among the following events:
• A: The sum is even.
• B: The sum is a multiple of 3.
• C: The sum is less than 4.
• D: The sum is greater than 11. Which pair of events are mutually exclusive?

• A. A and B
• B. A and C
• C. C and D
• D. B and D

Hint:

To test whether two events are mutually exclusive, find their intersection. If \[ A\cap B=\varnothing, \] then the events are mutually exclusive.

Answer 1

The Correct Option is C

Step 1: Determine the possible sums obtained from two dice. When two dice are thrown, the smallest possible sum is \[ 1+1=2, \] and the largest possible sum is \[ 6+6=12. \] Hence the possible sums are \[ \{2,3,4,5,6,7,8,9,10,11,12\}. \]

Step 2: Write the outcomes corresponding to each event. Event A: \[ A=\{2,4,6,8,10,12\}. \] Event B: \[ B=\{3,6,9,12\}. \] Event C: \[ C=\{2,3\}. \] Event D: \[ D=\{12\}. \]

Step 3: Check option (A): A and B. \[ A\cap B=\{6,12\}. \] Hence they are not mutually exclusive.

Step 4: Check option (B): A and C. \[ A\cap C=\{2\}. \] Hence they are not mutually exclusive.

Step 5: Check option (C): C and D. \[ C=\{2,3\}, \qquad D=\{12\}. \] \[ C\cap D=\varnothing. \] Therefore, these events are mutually exclusive.

Step 6: Check option (D): B and D. \[ B\cap D=\{12\}. \] Hence they are not mutually exclusive. Therefore, \[ {\text{C and D}} \] is the correct answer.