Find the probability that a number selected at random from 21 to 34 is a multiple of 3

1. Analyze the question

We need to find the probability of picking a multiple of 3 from the set of integers ranging from 21 to 34 (inclusive).

2. Explain the concept

Probability is defined as:

• Total outcomes: The count of all integers from 21 to 34.
• Favorable outcomes: The count of numbers within that range that are divisible by 3.

3. Show the solution

Step 1: Find the total number of outcomes. To find the number of integers in an inclusive range $[a, b]$, we use the formula: $(b - a) + 1$.

There are 14 possible numbers in total.

Step 2: Identify the multiples of 3 in the range. Let's list the numbers from 21 to 34 and identify the multiples of 3:

• 21 (3 × 7) - Yes
• 22
• 23
• 24 (3 × 8) - Yes
• 25
• 26
• 27 (3 × 9) - Yes
• 28
• 29
• 30 (3 × 10) - Yes
• 31
• 32
• 33 (3 × 11) - Yes
• 34

The multiples are {21, 24, 27, 30, 33}. Counting these, we have 5 favorable outcomes.

Step 3: Calculate the probability.

4. Identify the answer

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

5. Tip

When counting numbers in an inclusive range (e.g., "from X to Y"), always remember to add 1 to the difference $(Y - X)$. A common mistake students make is calculating $34 - 21 = 13$ and forgetting to include the starting number, which leads to an incorrect denominator!