Question

Set A consists of 35 consecutive integers. Quantity \(\mathrm{A}\) The probability of selecting an even number less than the median from set \(A\) Quantity \(\mathrm{B}\) The probability of selecting an odd number greater than the median from set \(A\) A Quantity A is greater. B Quantity B is greater. C The two quantities are equal. D The relationship cannot be determined from the information given.

Short answer

C The two quantities are equal.

Step-by-step solution

Understand the problem

The goal is to compare the probabilities of two events: selecting an even number less than the median from set A, and selecting an odd number greater than the median from set A.

Determine the median

Since set A consists of 35 consecutive integers, the median is the 18th number in the set because the median of an odd set of numbers is the middle number.

Count the even numbers less than the median

Before the median, there are 17 numbers. Since every second number is even, there are \(\frac{17}{2}\) even numbers, rounding down. This gives 8 even numbers.

Count the odd numbers greater than the median

After the median, there are also 17 numbers. Since every second number is odd, there are \(\frac{17}{2}\) odd numbers, rounding down. This gives 8 odd numbers.

Calculate the probabilities

For both quantities, the probability is the number we're interested in divided by the total number of numbers in set A. Therefore, both probabilities are \(\frac{8}{35}\).

Compare the quantities

Since both probabilities are \(\frac{8}{35}\), the two quantities are equal.

Key concepts

comparing probabilities

Comparing probabilities involves determining the likelihood that one event happens compared to another. In the given exercise, we compare two probabilities: the chance of picking an even number less than the median versus the chance of picking an odd number greater than the median.

- Finding both probabilities accurately ensures a proper comparison.
- If both probabilities are the same, the events have an equal chance of occurring.

In this case, after calculating, both events have probabilities of \(\frac{8}{35}\). This means each event is equally likely and directly leads to the conclusion that the two quantities are equal.

consecutive integers

A set of consecutive integers is a series of numbers where each number is one more than the previous number. For example, the numbers from 1 to 5 (1, 2, 3, 4, 5) are consecutive. In the exercise, set A is made up of 35 consecutive integers.

- The median of a set of consecutive integers is the middle number.
- For an odd number of consecutive integers, the median splits the set evenly.

In our example, the set of 35 consecutive integers has the 18th number as the median (since there are 17 numbers on either side), which is crucial in finding the even and odd numbers surrounding it for probabilities.

median of a set

The median is a measure of the central tendency of a set of numbers. For an odd number of elements, it's the middle number when all the elements are sorted in ascending order.

In a set of 35 consecutive integers, \(\text{median} = 18^{th} \text{number}\).
This means there are 17 numbers less than the median and 17 numbers greater than the median.

- For the probabilities in the exercise, knowing the median helps us determine the count of even and odd numbers on either side.

In the end, both even numbers less than the median and odd numbers greater than the median were counted as 8 each, guiding the final probability outcome equally.