Chapter 16: Problem 8

An integer \(N\) is chosen at random with \(1 \leq N \leq 100\). What is the probability that \(N\) is divisible by 11? That \(N>90\) ? That \(N \leq 3\) ? That \(N\) is a perfect square?

Short Answer

Expert verified

The probabilities are 0.09 for divisible by 11, 0.1 for greater than 90, 0.03 for less than or equal to 3, and 0.1 for perfect square.

Step by step solution

- Determine Total Possible Outcomes

Identify the range of possible outcomes. Since the integer can be any number from 1 to 100 inclusive, the total number of outcomes is 100.

- Find Numbers Divisible by 11

List the integers between 1 and 100 that are divisible by 11. These integers are: 11, 22, 33, 44, 55, 66, 77, 88, and 99. There are 9 such numbers.

- Calculate Probability for Divisibility by 11

The probability that an integer chosen at random is divisible by 11 is the number of favorable outcomes divided by the total number of possible outcomes. \[ P(N \text{ is divisible by 11}) = \frac{9}{100} = 0.09 \]

- Find Numbers Greater than 90

List the integers between 1 and 100 that are greater than 90. These integers are: 91, 92, 93, 94, 95, 96, 97, 98, 99, and 100. There are 10 such numbers.

- Calculate Probability for Numbers Greater than 90

The probability that an integer chosen at random is greater than 90 is the number of favorable outcomes divided by the total number of possible outcomes. \[ P(N > 90) = \frac{10}{100} = 0.1 \]

- Find Numbers Less Than or Equal to 3

List the integers between 1 and 100 that are less than or equal to 3. These integers are: 1, 2, and 3. There are 3 such numbers.

- Calculate Probability for Numbers Less Than or Equal to 3

The probability that an integer chosen at random is less than or equal to 3 is the number of favorable outcomes divided by the total number of possible outcomes. \[ P(N \leq 3) = \frac{3}{100} = 0.03 \]

- Find Perfect Squares

List the perfect squares between 1 and 100. These are: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. There are 10 such numbers.

- Calculate Probability for Perfect Squares

The probability that an integer chosen at random is a perfect square is the number of favorable outcomes divided by the total number of possible outcomes. \[ P(N \text{ is a perfect square}) = \frac{10}{100} = 0.1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Integer Selection

Random integer selection in probability refers to choosing a number from a specific range where each number has an equal chance of being selected. For example, in our exercise, the integers range from 1 to 100. Every number between these limits has an equal probability of being chosen.
To determine total possible outcomes, you count all the integers in the range. Here, since the numbers are from 1 to 100, the total outcomes are 100.
Understanding this concept is crucial because the total number of outcomes serves as the denominator in calculating probabilities.

Divisibility Rules

Divisibility rules help determine whether one number can be exactly divided by another without leaving a remainder. These rules are simple shortcuts that anyone can use without actually performing the division.
For example, a number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is a multiple of 11. Using this rule, the numbers between 1 and 100 that are divisible by 11 are 11, 22, 33, 44, 55, 66, 77, 88, and 99. We count 9 such numbers.
Divisibility rules simplify the process of identification, saving time and effort in probability calculations.

Probability Calculation

Probability is a measure of how likely an event is to occur. It is calculated as the number of favorable outcomes divided by the total number of possible outcomes. The formula is:
\[ P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]
Using our exercise, let's break down the calculations:

For numbers divisible by 11: Number of favorable outcomes = 9. Total possible outcomes = 100. \[ P(N \text{ is divisible by 11}) = \frac{9}{100} = 0.09 \]
For numbers greater than 90: Number of favorable outcomes = 10. \[ P(N > 90) = \frac{10}{100} = 0.1 \]
For numbers less than or equal to 3: Number of favorable outcomes = 3. \[ P(N \text{≤ 3}) = \frac{3}{100} = 0.03 \]

Probability helps us understand the likelihood of different outcomes.

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer with itself. For example, 1, 4, 9, 16, and so on, up to 100. To find perfect squares within a given range, list all integers whose squares fall within that range.
In our exercise, the perfect squares between 1 and 100 are 1 (1×1), 4 (2×2), 9 (3×3), 16 (4×4), 25 (5×5), 36 (6×6), 49 (7×7), 64 (8×8), 81 (9×9), and 100 (10×10).
There are a total of 10 perfect squares, making the probability of selecting a perfect square equal to:
\[ P(\text{Perfect Square}) = \frac{10}{100} = 0.1 \]Recognizing perfect squares quickly can assist in faster problem-solving and probability computation.