A number is selected at random between 20 and 30, both numbers inclusive. Find the probability that the number is a prime.

1. Analyze the question

We need to find the probability of selecting a prime number from the set of integers between 20 and 30, inclusive.

2. Explain the concept

• Sample Space ($n(S)$): This is the total number of possible outcomes. Since the range is 20 to 30 inclusive, we count every integer from 20 up to 30.
• Prime Number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
• Probability Formula: $P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

3. Show the solution

Step 1: Determine the total number of outcomes ($n(S)$) The numbers are: 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30. Counting these, we have: $n(S) = 11$

Step 2: Identify the prime numbers in the set ($n(E)$) Let's check each number:

• 20: Not prime (divisible by 2, 4, 5, 10)
• 21: Not prime (divisible by 3, 7)
• 22: Not prime (divisible by 2, 11)
• 23: Prime
• 24: Not prime (divisible by 2, 3, 4, 6, 8, 12)
• 25: Not prime (divisible by 5)
• 26: Not prime (divisible by 2, 13)
• 27: Not prime (divisible by 3, 9)
• 28: Not prime (divisible by 2, 4, 7, 14)
• 29: Prime
• 30: Not prime (divisible by 2, 3, 5, 6, 10, 15)

The prime numbers are {23, 29}. So, $n(E) = 2$.

Step 3: Calculate the probability $P(\text{Prime}) = \frac{n(E)}{n(S)} = \frac{2}{11}$

4. Identify the answer

Comparing our result to the given options, the correct choice is A.

5. Tip

When counting numbers in an "inclusive" range (from $a$ to $b$), use the formula $(b - a) + 1$. For this question: $(30 - 20) + 1 = 10 + 1 = 11$. This ensures you don't accidentally miss one of the endpoints!