Cards numbered 1 to 30 are put in a bag. A card is drawn at random from this bag. Find the probability that the number on the drawn card is:
(i) not divisible by 3.
(ii) a prime number greater than 7.
(iii) not a perfect square number.

Answer

To solve the problem step by step, we will calculate the required probabilities for each part of the question.

**Total number of cards:**
The cards are numbered from 1 to 30, so the total number of cards is 30.

### Part (i): Probability that the number on the drawn card is not divisible by 3.

1. **Find numbers divisible by 3:**
The numbers divisible by 3 between 1 and 30 are:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
There are a total of 10 numbers.

2. **Calculate probability of being divisible by 3:**
Probability (P) of drawing a card that is divisible by 3 is given by:
\[ P(\text{divisible by 3}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{10}{30} = \frac{1}{3} \]

3. **Calculate probability of not being divisible by 3:**
\[ P(\text{not divisible by 3}) = 1 - P(\text{divisible by 3}) = 1 - \frac{1}{3} = \frac{2}{3} \]

### Part (ii): Probability that the number on the drawn card is a prime number greater than 7.

1. **Identify prime numbers greater than 7:**
The prime numbers greater than 7 and less than or equal to 30 are:
11, 13, 17, 19, 23, 29.
There are a total of 6 prime numbers.

2. **Calculate probability of drawing a prime number greater than 7:**
\[ P(\text{prime number > 7}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{30} = \frac{1}{5} \]

### Part (iii): Probability that the number on the drawn card is not a perfect square number.

1. **Identify perfect square numbers between 1 and 30:**
The perfect squares are:
1 (1²), 4 (2²), 9 (3²), 16 (4²), 25 (5²).
There are a total of 5 perfect squares.

2. **Calculate probability of drawing a perfect square number:**
\[ P(\text{perfect square}) = \frac{5}{30} = \frac{1}{6} \]

3. **Calculate probability of not drawing a perfect square number:**
\[ P(\text{not a perfect square}) = 1 - P(\text{perfect square}) = 1 - \frac{1}{6} = \frac{5}{6} \]

### Final Answers:
- (i) Probability that the number is not divisible by 3: \(\frac{2}{3}\)
- (ii) Probability that the number is a prime number greater than 7: \(\frac{1}{5}\)
- (iii) Probability that the number is not a perfect square number: \(\frac{5}{6}\)