Question

One integer will be randomly selected from the integers 11 to 60, inclusive. What is the probability that the selected integer will be a perfect square or a perfect cube?

• A. 0.1
• B. 0.125
• C. 0.16
• D. 0.5
• E. 0.9

Hint:

When calculating the number of integers in an inclusive range (from a to b), the formula is \(b - a + 1\). A common mistake is to just subtract (\(b-a\)).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a probability problem. We need to find the number of favorable outcomes (integers that are perfect squares or perfect cubes in the given range) and divide it by the total number of possible outcomes.
Step 2: Key Formula or Approach:
\[ P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \]
For "A or B" events, \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\). We must check for any overlap (numbers that are both a perfect square and a perfect cube).
Step 3: Detailed Explanation:
1. Find the total number of outcomes.
The integers are from 11 to 60, inclusive.
Total numbers = \(60 - 11 + 1 = 50\).
2. Find the number of favorable outcomes.
\textit{Perfect Squares} in the range [11, 60]:
\(4^2 = 16\), \(5^2 = 25\), \(6^2 = 36\), \(7^2 = 49\). (\(3^2=9\) is too small, \(8^2=64\) is too large). There are 4 perfect squares.
\textit{Perfect Cubes} in the range [11, 60]:
\(3^3 = 27\). (\(2^3=8\) is too small, \(4^3=64\) is too large). There is 1 perfect cube.
3. Check for overlap.
Is any number in our list both a square and a cube? The lists are \{16, 25, 36, 49\} and \{27\}. There is no overlap.
4. Calculate the total number of favorable outcomes.
Total favorable outcomes = (Number of squares) + (Number of cubes) = \(4 + 1 = 5\).
5. Calculate the probability.
\[ P(\text{square or cube}) = \frac{5}{50} = \frac{1}{10} = 0.1 \]
Step 4: Final Answer:
The probability is 0.1.