Find the probability that a number picked at random from the set(43, 44, 45, ..., 60) is a prime number.

1. Analyze the Question

We need to find the probability of picking a prime number from the set of integers ranging from 43 to 60 inclusive. Probability is defined as:

2. Explain the Concept

• Total outcomes: To find the number of integers in a range $[a, b]$, use the formula: $(b - a) + 1$.
• Prime numbers: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. We need to carefully check each number in the set to see if it is divisible by any smaller prime numbers (like 2, 3, 5, or 7).

3. Show the Solution

Step 1: Find the total number of elements in the set. The set is $\{43, 44, 45, \dots, 60\}$. Total count = $(60 - 43) + 1 = 17 + 1 = 18$.

Step 2: Identify the prime numbers in the set. Let's list the numbers and check them:

• 43: Prime (Yes)
• 44: Even (No)
• 45: Divisible by 5 (No)
• 46: Even (No)
• 47: Prime (Yes)
• 48: Even (No)
• 49: $7 \times 7$ (No)
• 50: Even (No)
• 51: $3 \times 17$ (No)
• 52: Even (No)
• 53: Prime (Yes)
• 54: Even (No)
• 55: Divisible by 5 (No)
• 56: Even (No)
• 57: $3 \times 19$ (No)
• 58: Even (No)
• 59: Prime (Yes)
• 60: Even (No)

The prime numbers are $\{43, 47, 53, 59\}$. Number of favorable outcomes = 4.

Step 3: Calculate the probability.

Simplify the fraction by dividing both numerator and denominator by 2:

4. Identify the Answer

Comparing our result to the given options, we find that it matches option C.

5. Exam Tip

When counting numbers in a range, students often forget to add the "+ 1". Always remember: if you are counting from 1 to 10, there are $(10 - 1) + 1 = 10$ numbers, not 9! Also, for numbers like 51 and 57, always check for divisibility by 3 by adding the digits (e.g., $5+1=6$, which is divisible by 3, so 51 is not prime).