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### Problem Summary:
You are given a sample space \( S = \{ x : 1 \leq x < 20, \, x \text{ is an integer} \} \). A number is taken at random from \( S \). You need to find the probability that the number is:
- (a) A perfect square.
- (b) A prime number.

### Step 1: Determine the Sample Space
The sample space \( S \) consists of all integers from 1 to 19, inclusive. Therefore, the total number of elements in \( S \) is:
\[
n(S) = 19
\]

### Step 2: (a) Find the Probability that the Number is a Perfect Square
A perfect square is a number that can be expressed as \( k^2 \) where \( k \) is an integer. 

In the sample space \( S \), the perfect squares are:
\[
1^2 = 1, \, 2^2 = 4, \, 3^2 = 9, \, 4^2 = 16
\]
So, the perfect squares within \( S \) are \( \{1, 4, 9, 16\} \).

The number of perfect squares:
\[
n(\text{perfect squares}) = 4
\]

The probability that the number is a perfect square:
\[
P(\text{perfect square}) = \frac{n(\text{perfect squares})}{n(S)} = \frac{4}{19}
\]

### Step 3: (b) Find the Probability that the Number is a Prime Number
A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

In the sample space \( S \), the prime numbers are:
\[
2, 3, 5, 7, 11, 13, 17, 19
\]
So, the prime numbers within \( S \) are \( \{2, 3, 5, 7, 11, 13, 17, 19\} \).

The number of prime numbers:
\[
n(\text{prime numbers}) = 8
\]

The probability that the number is a prime number:
\[
P(\text{prime number}) = \frac{n(\text{prime numbers})}{n(S)} = \frac{8}{19}
\]

### Final Answer:
- (a) The probability that the number is a perfect square is \( \frac{4}{19} \).
- (b) The probability that the number is a prime number is \( \frac{8}{19} \).

Would you like more details or have any questions?

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Here are five related questions:
1. What is the probability of selecting an even number from the sample space?
2. How would the probabilities change if the sample space were \( S = \{ x : 1 \leq x \leq 20, \, x \text{ is an integer} \)?
3. What is the probability of selecting a number that is neither a perfect square nor a prime number?
4. How do you calculate the probability of selecting a number divisible by 3 from the sample space?
5. What is the probability of selecting a number greater than 10 from the sample space?

**Tip:** When calculating probabilities, always ensure that the total number of possible outcomes is correctly identified before counting the favorable outcomes.

Given the sample space S = {x : 1 ≤ x < 20, x is an integer}. A number is taken at random from S. Find the probability that the number is (a) a perfect square, (b) a prime number.

Learn how to find the probability of selecting a perfect square or a prime number from a given sample space S = {x : 1 ≤ x < 20, x is an integer}. This problem involves key probability concepts, perfect squares, and prime numbers, suitable for Grades 7-9.

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