To solve the problem, we will find the probability of drawing a card with a specific number based on the criteria given.
### Step-by-step Solution:
**Total Cards:**
1. The cards are numbered from 2 to 151.
2. Therefore, the total number of cards = 151 - 2 + 1 = 150.
**(i) Probability of drawing a prime number less than 75:**
1. **Identify Prime Numbers Less Than 75:**
- The prime numbers less than 75 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71.
- Count these prime numbers:
- There are 21 prime numbers less than 75.
2. **Calculate Probability:**
- The probability formula is given by:
\[
\text{Probability} = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}
\]
- Here, the number of favourable outcomes (prime numbers less than 75) = 21.
- Total number of outcomes (total cards) = 150.
- Thus, the probability of drawing a prime number less than 75 is:
\[
\text{Probability} = \frac{21}{150} = \frac{7}{50}.
\]
**(ii) Probability of drawing an odd number:**
1. **Identify Total Odd Numbers:**
- The numbers from 2 to 151 consist of both odd and even numbers.
- The odd numbers in this range are:
3, 5, 7, 9, 11, ..., 151.
- The first odd number is 3 and the last is 151.
- The odd numbers form an arithmetic sequence where:
- First term (a) = 3,
- Last term (l) = 151,
- Common difference (d) = 2.
- To find the number of terms (n) in this sequence, we can use the formula:
\[
n = \frac{l - a}{d} + 1 = \frac{151 - 3}{2} + 1 = \frac{148}{2} + 1 = 74 + 1 = 75.
\]
2. **Calculate Probability:**
- The probability of drawing an odd number is:
\[
\text{Probability} = \frac{\text{Number of odd numbers}}{\text{Total number of outcomes}} = \frac{75}{150} = \frac{1}{2}.
\]
### Final Answers:
- (i) The probability of drawing a prime number less than 75 is \(\frac{7}{50}\).
- (ii) The probability of drawing an odd number is \(\frac{1}{2}\).
