Question

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

P(N divisible by 11) = 0.09, P(N>90) = 0.10, P(N≤3) = 0.03, P(N is a perfect square) = 0.10.

Step-by-step solution

Determine the total number of possible integers

The integers range from 1 to 100, inclusive. Therefore, the total number of possible integers is 100.

Find numbers divisible by 11

The integers divisible by 11 between 1 and 100 are 11, 22, 33, 44, 55, 66, 77, 88, and 99. Therefore, there are 9 such numbers.

Calculate the probability of divisibility by 11

The probability that a randomly chosen integer is divisible by 11 is given by the ratio of the count of such numbers to the total count: \[P(N \text{ is divisible by } 11) = \frac{9}{100} = 0.09\]

Find numbers greater than 90

The integers greater than 90 are 91, 92, 93, 94, 95, 96, 97, 98, 99, and 100. Thus, there are 10 such numbers.

Calculate the probability of being greater than 90

The probability that a randomly chosen integer is greater than 90 is: \[P(N > 90) = \frac{10}{100} = 0.10\]

Find numbers less than or equal to 3

The integers less than or equal to 3 are 1, 2, and 3. Therefore, there are 3 such numbers.

Calculate the probability of being less than or equal to 3

The probability that a randomly chosen integer is less than or equal to 3 is: \[P(N \text{ is } \leq 3) = \frac{3}{100} = 0.03\]

Find perfect squares between 1 and 100

The perfect squares between 1 and 100 are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. Hence, there are 10 perfect squares.

Calculate the probability of being a perfect square

The probability that a randomly chosen integer is a perfect square is: \[P(N \text{ is a perfect square}) = \frac{10}{100} = 0.10\]

Key concepts

divisibility

In mathematics, divisibility refers to the ability of one integer to be divided by another without leaving a remainder. For an integer to be divisible by another, the division of these two numbers should result in a whole number.
For example, 15 is divisible by 5 because when you divide 15 by 5, you get 3, which is a whole number.
Divisibility plays a significant role in various fields, including number theory, where it helps in determining prime numbers and in solving equations.
In our exercise, the numbers from 1 to 100 divisible by 11 are: 11, 22, 33, 44, 55, 66, 77, 88, and 99.
To find if a number is divisible by 11 among any set of integers, observe if the numbers perfectly divide without a remainder.

random integers

Random integers are numbers chosen in such a way that every possible number in a given set has an equal chance of being selected. This process is crucial in probability and statistics for creating unbiased samples.
For the given exercise, we look at integers randomly chosen from 1 to 100. There is no bias in selection, meaning the integer could just as likely be 50 as it could be 1.
This randomness helps in calculating the probability of certain conditions, like being divisible by 11 or greater than 90.
The key aspect of randomness is that it assures every number in the defined range has equal opportunity to be selected, ensuring a fair analysis in probability calculations.

perfect squares

Perfect squares are numbers that result from squaring a whole number. For example, 25 is a perfect square because it is the result of 5 times 5.
Perfect squares are important in various areas of mathematics, including algebra and geometry. They simplify complex equations and can represent areas of squares in geometric problems.
In our exercise, the perfect squares between 1 and 100 are: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
Spotting perfect squares within a range helps in solving probability problems where the condition involves numbers that are perfect squares.
It’s essential to know that the intervals between perfect squares grow larger as the numbers get bigger, which can sometimes make identifying them a bit more challenging.

mathematical probability calculations

Mathematical probability is the measure of the likelihood that an event will occur. In probability theory, we often use calculations to determine this likelihood.
To find the probability of a specific event, we divide the number of successful outcomes by the total number of possible outcomes.
In the given exercise, we calculate the probability of different scenarios such as a number being divisible by 11 or greater than 90.
The formula used for these calculations is simple: \ \[ P(Event) = \frac{Number \ of \ Successful \ Outcomes}{Total \ Number \ of \ Possible \ Outcomes} \ \]
For example, the probability that a number is greater than 90 is given by: \ \[ \frac{10}{100} = 0.10 \] Where 10 is the count of numbers greater than 90.
Understanding this basic probability formula helps solve a variety of problems involving chance and likelihood.