7. Probability 5 (Math Set 11, Batch24)

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na game, balls are labeled with perfect squares less than 1,000,001 and placed in a bag. If a ball is drawn at random, what is the probability that the number on the all is also a perfect cube?

te: Round off your answer to 3 decimal places. For example if the answer is 0.5 put 0.500 and if the answer is 0.5447 then put 0.5

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To solve this problem, we first need to identify all the numbers that are perfect squares less than 1,000,001 and then determine which of those numbers are also perfect cubes.

### Step 1: Finding perfect squares
The largest integer \( n \) such that \( n^2 \) is less than 1,000,001 is found by solving:
\[ n^2 < 1,000,001 \]
\[ n < \sqrt{1,000,001} \approx 1000 \]
Thus, the perfect squares less than 1,000,001 are \( 1^2, 2^2, 3^2, \ldots, 1000^2 \).

So, there are 1000 perfect squares.

### Step 2: Finding perfect cubes
Similarly, the largest integer \( m \) such that \( m^3 \) is less than 1,000,001 is found by solving:
\[ m^3 < 1,000,001 \]
\[ m < \sqrt[3]{1,000,001} \approx 100 \]
Thus, the perfect cubes less than 1,000,001 are \( 1^3, 2^3, 3^3, \ldots, 100^3 \).

So, there are 100 perfect cubes.

### Step 3: Finding numbers that are both perfect squares and perfect cubes
A number that is both a perfect square and a perfect cube is a perfect sixth power. The largest integer \( k \) such that \( k^6 \) is less than 1,000,001 is found by solving:
\[ k^6 < 1,000,001 \]
\[ k < \sqrt[6]{1,000,001} \approx 10 \]
Thus, the perfect sixth powers less than 1,000,001 are \( 1^6, 2^6, 3^6, \ldots, 10^6 \).

So, there are 10 numbers that are both perfect squares and perfect cubes.

### Step 4: Calculating the probability
The probability that a randomly drawn ball has a number that is both a perfect square and a perfect cube is:
\[ P = \frac{\text{Number of perfect sixth powers}}{\text{Number of perfect squares}} = \frac{10}{1000} = \frac{1}{100} = 0.01 \]

Thus, the probability is \( 0.010 \) when rounded to three decimal places.

So, the answer is \( 0.010 \).

Would you like more details or have any questions?

Here are 5 more questions you might be interested in:
1. How do you determine if a number is a perfect power?
2. What is the significance of identifying numbers that are both perfect squares and perfect cubes?
3. Can you explain the process of finding the largest integer within a certain power limit?
4. How does probability theory apply to games of chance like drawing balls from a bag?
5. What are some real-world applications of probability calculations involving perfect powers?

Tip: When working with large numbers and exponents, use a calculator or computer software to ensure accuracy in your calculations.

Explore the probability of drawing a perfect cube number from a collection of perfect squares less than 1,000,001. Understand the process of identifying perfect squares, perfect cubes, and their intersection. Calculate the probability step-by-step with detailed explanations and examples. Suitable for advanced high school students.

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