A die is thrown once. If probabililty of an event X is denoted by P(X), find:
a. (i) P (even number)
(ii) P(a number `ge3`)
) (iii) P (a number between 2 and 5)
b. (i) P (a number `ge4`)
(ii) P(a number `lt7`)
(iii) P (a number `gt6`).

Answer

To solve the problem, we will analyze the outcomes of throwing a die and calculate the probabilities for the specified events.

### Step-by-Step Solution:

1. **Identify the Sample Space**:
The sample space when a die is thrown consists of the numbers {1, 2, 3, 4, 5, 6}. This gives us a total of 6 possible outcomes.

2. **Calculate P(even number)**:
- The even numbers in the sample space are {2, 4, 6}.
- Number of favorable outcomes = 3 (even numbers).
- Total outcomes = 6.
- Probability, P(even number) = Number of favorable outcomes / Total outcomes = 3 / 6 = 1 / 2.

3. **Calculate P(a number ≥ 3)**:
- The numbers greater than or equal to 3 in the sample space are {3, 4, 5, 6}.
- Number of favorable outcomes = 4.
- Total outcomes = 6.
- Probability, P(a number ≥ 3) = 4 / 6 = 2 / 3.

4. **Calculate P(a number between 2 and 5)**:
- The numbers between 2 and 5 (exclusive) are {3, 4}.
- Number of favorable outcomes = 2.
- Total outcomes = 6.
- Probability, P(a number between 2 and 5) = 2 / 6 = 1 / 3.

5. **Calculate P(a number ≥ 4)**:
- The numbers greater than or equal to 4 are {4, 5, 6}.
- Number of favorable outcomes = 3.
- Total outcomes = 6.
- Probability, P(a number ≥ 4) = 3 / 6 = 1 / 2.

6. **Calculate P(a number < 7)**:
- All numbers in the sample space {1, 2, 3, 4, 5, 6} are less than 7.
- Number of favorable outcomes = 6.
- Total outcomes = 6.
- Probability, P(a number < 7) = 6 / 6 = 1.

7. **Calculate P(a number > 6)**:
- There are no numbers in the sample space that are greater than 6.
- Number of favorable outcomes = 0.
- Total outcomes = 6.
- Probability, P(a number > 6) = 0 / 6 = 0.

### Final Answers:
a.
(i) P(even number) = 1/2
(ii) P(a number ≥ 3) = 2/3
(iii) P(a number between 2 and 5) = 1/3

b.
(i) P(a number ≥ 4) = 1/2
(ii) P(a number < 7) = 1
(iii) P(a number > 6) = 0