An unbiased die is thrown. What is the probability of getting:
`(i)` an even number
`(ii)` a multiple of `3`
`(iii`) an even number or a multiple of `3`
`(iv)` an even number and a multiple of `3`
`(v)` a number `3` or `4`
`(vi)` an odd number
`(vii)` a number less than `5`
`(viii)` a number greater than `3`
`(ix)` a number between `3` and `6`

Answer

To solve the problem of finding the probabilities of various outcomes when an unbiased die is thrown, we will follow these steps:

### Step-by-Step Solution:

1. **Total Outcomes**:
- When a die is thrown, the total number of possible outcomes is 6 (1, 2, 3, 4, 5, 6).

2. **(i) Probability of getting an even number**:
- Even numbers on a die: 2, 4, 6.
- Favorable outcomes = 3 (even numbers).
- Probability = Favorable outcomes / Total outcomes = 3 / 6 = 1 / 2.

3. **(ii) Probability of getting a multiple of 3**:
- Multiples of 3 on a die: 3, 6.
- Favorable outcomes = 2 (multiples of 3).
- Probability = 2 / 6 = 1 / 3.

4. **(iii) Probability of getting an even number or a multiple of 3**:
- Even numbers: 2, 4, 6.
- Multiples of 3: 3, 6.
- Combined favorable outcomes: 2, 4, 6, 3 (total unique outcomes = 4).
- Probability = 4 / 6 = 2 / 3.

5. **(iv) Probability of getting an even number and a multiple of 3**:
- The only number that is both even and a multiple of 3 is 6.
- Favorable outcomes = 1.
- Probability = 1 / 6.

6. **(v) Probability of getting a number 3 or 4**:
- Favorable outcomes = 3, 4 (total = 2).
- Probability = 2 / 6 = 1 / 3.

7. **(vi) Probability of getting an odd number**:
- Odd numbers on a die: 1, 3, 5.
- Favorable outcomes = 3.
- Probability = 3 / 6 = 1 / 2.

8. **(vii) Probability of getting a number less than 5**:
- Numbers less than 5: 1, 2, 3, 4.
- Favorable outcomes = 4.
- Probability = 4 / 6 = 2 / 3.

9. **(viii) Probability of getting a number greater than 3**:
- Numbers greater than 3: 4, 5, 6.
- Favorable outcomes = 3.
- Probability = 3 / 6 = 1 / 2.

10. **(ix) Probability of getting a number between 3 and 6**:
- Numbers between 3 and 6: 4, 5.
- Favorable outcomes = 2.
- Probability = 2 / 6 = 1 / 3.

### Summary of Probabilities:
- (i) Even number: 1/2
- (ii) Multiple of 3: 1/3
- (iii) Even number or multiple of 3: 2/3
- (iv) Even number and multiple of 3: 1/6
- (v) Number 3 or 4: 1/3
- (vi) Odd number: 1/2
- (vii) Number less than 5: 2/3
- (viii) Number greater than 3: 1/2
- (ix) Number between 3 and 6: 1/3