To solve the problem step by step, we will calculate the probability for each case as follows:
### Total Tokens
The total number of tokens in the bag is 100 (numbered from 1 to 100).
### (i) Probability of drawing an even number
- **Even numbers from 1 to 100**: 2, 4, 6, ..., 100
- **Count of even numbers**: There are 50 even numbers (2, 4, 6, ..., 100).
- **Probability**:
\[
P(\text{Even}) = \frac{\text{Number of even numbers}}{\text{Total numbers}} = \frac{50}{100} = \frac{1}{2}
\]
### (ii) Probability of drawing an odd number
- **Odd numbers from 1 to 100**: 1, 3, 5, ..., 99
- **Count of odd numbers**: There are also 50 odd numbers (1, 3, 5, ..., 99).
- **Probability**:
\[
P(\text{Odd}) = \frac{\text{Number of odd numbers}}{\text{Total numbers}} = \frac{50}{100} = \frac{1}{2}
\]
### (iii) Probability of drawing a multiple of 3
- **Multiples of 3 from 1 to 100**: 3, 6, 9, ..., 99
- **Count of multiples of 3**: The multiples of 3 up to 100 are 3, 6, 9, ..., 99, which gives us 33 multiples.
- **Probability**:
\[
P(\text{Multiple of 3}) = \frac{33}{100}
\]
### (iv) Probability of drawing a multiple of 5
- **Multiples of 5 from 1 to 100**: 5, 10, 15, ..., 100
- **Count of multiples of 5**: The multiples of 5 up to 100 are 5, 10, 15, ..., 100, which gives us 20 multiples.
- **Probability**:
\[
P(\text{Multiple of 5}) = \frac{20}{100} = \frac{1}{5}
\]
### (v) Probability of drawing a multiple of both 3 and 5
- **Multiples of 15 from 1 to 100**: 15, 30, 45, 60, 75, 90
- **Count of multiples of 15**: There are 6 multiples of 15.
- **Probability**:
\[
P(\text{Multiple of 3 and 5}) = \frac{6}{100} = \frac{3}{50}
\]
### (vi) Probability of drawing a multiple of 3 or 5
Using the formula for the union of two events:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
Where:
- \(P(A)\) = Probability of multiple of 3 = \(\frac{33}{100}\)
- \(P(B)\) = Probability of multiple of 5 = \(\frac{20}{100}\)
- \(P(A \cap B)\) = Probability of multiple of both 3 and 5 = \(\frac{6}{100}\)
Calculating:
\[
P(\text{Multiple of 3 or 5}) = \frac{33}{100} + \frac{20}{100} - \frac{6}{100} = \frac{47}{100}
\]
### (vii) Probability of drawing a number less than 20
- **Numbers less than 20**: 1, 2, 3, ..., 19
- **Count of numbers less than 20**: There are 19 such numbers.
- **Probability**:
\[
P(\text{Less than 20}) = \frac{19}{100}
\]
### (viii) Probability of drawing a number greater than 70
- **Numbers greater than 70**: 71, 72, ..., 100
- **Count of numbers greater than 70**: There are 30 such numbers.
- **Probability**:
\[
P(\text{Greater than 70}) = \frac{30}{100} = \frac{3}{10}
\]
### Summary of Probabilities
1. \(P(\text{Even}) = \frac{1}{2}\)
2. \(P(\text{Odd}) = \frac{1}{2}\)
3. \(P(\text{Multiple of 3}) = \frac{33}{100}\)
4. \(P(\text{Multiple of 5}) = \frac{1}{5}\)
5. \(P(\text{Multiple of 3 and 5}) = \frac{3}{50}\)
6. \(P(\text{Multiple of 3 or 5}) = \frac{47}{100}\)
7. \(P(\text{Less than 20}) = \frac{19}{100}\)
8. \(P(\text{Greater than 70}) = \frac{3}{10}\)
