Two dice are rolled Let A,B and C be the events of getting a sum 2, a sum 3 and a sum 4 respectively.
a. (i) Is event A simple?
(ii) Is event B simple?
(iii) Is event C compound?
b. Are events A and B mutually exclusive?

Answer

To solve the problem step by step, we will analyze the events A, B, and C based on the outcomes when two dice are rolled.

### Step-by-Step Solution:

**Step 1: Identify the Sample Space**
When two dice are rolled, the total number of possible outcomes is \(6 \times 6 = 36\). The sample space can be represented as:
\[ S = \{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)\} \]

**Step 2: Define Events A, B, and C**
- Event A (sum = 2): The only outcome is \((1,1)\).
- Event B (sum = 3): The possible outcomes are \((1,2)\) and \((2,1)\).
- Event C (sum = 4): The possible outcomes are \((1,3)\), \((2,2)\), and \((3,1)\).

**Step 3: Determine if Event A is Simple**
A simple event has only one outcome. Since event A has only one outcome \((1,1)\), we conclude:
- **Answer**: Yes, event A is a simple event.

**Step 4: Determine if Event B is Simple**
Event B has two outcomes: \((1,2)\) and \((2,1)\). Since it has more than one outcome, we conclude:
- **Answer**: No, event B is not a simple event; it is a compound event.

**Step 5: Determine if Event C is Compound**
Event C has three outcomes: \((1,3)\), \((2,2)\), and \((3,1)\). Since it has more than one outcome, we conclude:
- **Answer**: Yes, event C is a compound event.

**Step 6: Check if Events A and B are Mutually Exclusive**
Two events are mutually exclusive if they cannot occur at the same time, meaning their intersection is empty.
- Event A has the outcome \((1,1)\).
- Event B has outcomes \((1,2)\) and \((2,1)\).
Since there are no common outcomes between event A and event B, we conclude:
- **Answer**: Yes, events A and B are mutually exclusive.

### Summary of Answers:
a.
(i) Yes, event A is a simple event.
(ii) No, event B is not a simple event; it is a compound event.
(iii) Yes, event C is a compound event.

b. Yes, events A and B are mutually exclusive.