# Chapter 15 – Probability and Statistics

Probability and statistics are crucial twenty-first-century mathematical tools! Selected topics have been incorporated and mandated into many state frameworks and mathematics curricula. The study of probability and statistics has been around for a long time. An important statistical test (t test) was developed by William Gossett of the Guinness Brewery at the turn of the century to keep hops and barley in the best condition. At the same time, John Dewey was using a “scientific method” that utilized no statistical processes. His well-accepted educational philosophies that were tested in his laboratory school were the result of a sample size of 10 nonrandomly chosen students. His scientific method used 80 years ago would not be accepted as a valid research practice in master’s or doctoral programs today. Maybe that says statistics is new. Statistics has received little to no emphasis in the secondary curriculum until relatively recently. That is changing, as evidenced by the number of texts that now include statistics and probability topics.

## Exercises

### Exercise 15.1

- View
*USA Today*(www.usatoday.com)*or ESPN*(www.espn.com) on the Internet. Find a poll on one of these sites. Place your vote on one of their online polls. How are the results displayed? Do you feel this graphical representation is the best way to display the results? Can you enter your vote more than once? Should you be worried about skewed results? - What would you say to a student who comes from a limited-income environment who talks about spending money on the lottery?
- Develop a lesson that would teach a “general mathematics” class about the lack of wisdom of playing the lottery and similar games of chance for money.
- Odds are calculated differently than probabilities. Outline a lesson plan that you would use to help students understand the differences.
- Read
*USA Today*. React to at least two of the graphs shown. How were the data gathered? How many were in the sample? Is any statistical information given? - Read at least a part of a book that deals with the potential of telling lies with statistics. Report on at least two new revelations for you.

### Exercise 15.2

- Is it possible to insert statistical concepts into the secondary curriculum at multiple levels? Defend your position.
- Where do the Common Core Standards expect you to integrate probability and statistics into the curriculum? What about the elementary curriculum?

What would you do if you were told that the only formal statistical instruction that is to be done in your school is to be limited to the AP class?

## Problem Solving Challenges

- Your teacher displays a monthly calendar in your math class indicating birthdays of you and your fellow students. Your math class contains 25 students. What is the probability that three or more students in your math class were born in the same month?

**Hint: **Think of 13 students. What is the chance that 2 students will have the same birth month?

## Answer

Answer/solution: 1.

In a class of 24 students, it is possible that exactly two students would have been born in each of the 12 months. For 25 or more students, it is certain that there would be at least one month with three or more. Therefore, the probability is one.

- The arithmetic mean of a set of nine different positive integers is 123456789. Each number in the set contains a different number of digits with the greatest value being a nine-digit number. Find the value of each of the nine numbers.

**Hint: **The problem states that each number in the set contains a different number of digits but it did not state that the numbers cannot all have the same digit.

## Answer

Answer/solution: 9, 99, 999, 9999, 99999, 999999, 9999999, 99999999, 999999999.

Since the mean of the numbers is 123456789, the sum of the numbers must be 9 x 123456789 = 1111111101. We know that we will have a one-digit, two-digit, three-digit etc. number in the set. By trial and error, first try the smallest values: 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111 = 123456789, which is not even ten digits. Trying the greatest values gives us 1111111101.

## Additional Learning Activities

There is no Additional Learning Activities for this chapter.