A die is thrown, find the probability of following events:
(i) A prime number will appear,
(ii) A number greater than or equal to 3 will appear,
(iii) A number less than or equal to one will appear,
(iv) A number more than 6 will appear,
(v) A number less than 6 will appear.

Explanation

To solve the problem, we will calculate the probability for each of the specified events when a die is thrown.

A standard die has 6 faces, numbered from 1 to 6. The total number of outcomes when a die is thrown is 6.

### (i) Probability of a prime number appearing

**Step 1:** Identify the prime numbers on a die.
The prime numbers between 1 and 6 are 2, 3, and 5.
So, the prime numbers are: {2, 3, 5}.

**Step 2:** Count the number of favorable outcomes.
The number of prime numbers = 3.

**Step 3:** Calculate the probability.
Probability (A) = Number of favorable outcomes / Total outcomes = 3 / 6 = 1 / 2.

### (ii) Probability of a number greater than or equal to 3 appearing

**Step 1:** Identify the numbers greater than or equal to 3.
The numbers are: {3, 4, 5, 6}.

**Step 2:** Count the number of favorable outcomes.
The number of favorable outcomes = 4.

**Step 3:** Calculate the probability.
Probability (B) = Number of favorable outcomes / Total outcomes = 4 / 6 = 2 / 3.

### (iii) Probability of a number less than or equal to 1 appearing

**Step 1:** Identify the numbers less than or equal to 1.
The only number is: {1}.

**Step 2:** Count the number of favorable outcomes.
The number of favorable outcomes = 1.

**Step 3:** Calculate the probability.
Probability (C) = Number of favorable outcomes / Total outcomes = 1 / 6.

### (iv) Probability of a number more than 6 appearing

**Step 1:** Identify the numbers more than 6.
There are no numbers on a die that are more than 6.

**Step 2:** Count the number of favorable outcomes.
The number of favorable outcomes = 0.

**Step 3:** Calculate the probability.
Probability (D) = Number of favorable outcomes / Total outcomes = 0 / 6 = 0.

### (v) Probability of a number less than 6 appearing

**Step 1:** Identify the numbers less than 6.
The numbers are: {1, 2, 3, 4, 5}.

**Step 2:** Count the number of favorable outcomes.
The number of favorable outcomes = 5.

**Step 3:** Calculate the probability.
Probability (E) = Number of favorable outcomes / Total outcomes = 5 / 6.

### Summary of Probabilities:
- (i) Probability of a prime number = 1/2
- (ii) Probability of a number ≥ 3 = 2/3
- (iii) Probability of a number ≤ 1 = 1/6
- (iv) Probability of a number > 6 = 0
- (v) Probability of a number < 6 = 5/6