# Fair and Unfair Worksheets

The concept of mathematical fairness can be quantified in simple terms. For an event to fair it must match simple probability meaning that you must have an equal chance of both winning and losing. Many different games of chance tilt that advantage where you have a greater chance of losing than winning. Being able to understand if you are playing a fair game, is important towards understanding if it is worth your time or money. This is a series of worksheets that run students through a wide variety of ways to calculate probabilities and then we help students interpret the data to decide what is fair or as close to equal as possible.

### Aligned Standard: HSS-MD.B.6

- Choosing Batting Order Step-by-step Lesson- Which method is fair for choosing the batting order?
- Guided Lesson - We try to determine if a dice game and two spinners are fair?
- Guided Lesson Explanation - This reminds me that judges have a really tough job at any level.
- Practice Worksheet - There are some fun ones in here. Banana eating, buying an audio systems, and team captains are all included.
- Matching Worksheet - There are only two examples in here. This is supposed to start the discussion on fairness.
- Combinations Five Pack - I always like these types of problems. Many times they are referred to as brain teasers.
- Empirical Probability Five Pack - These are a refresher for you to tie into this standard.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

Are the games and methods of choosing turns fair?

- Homework 1 - 3 friends are playing with dice. They decide to roll a die. If it lands on 1 then you win $10. If it lands on 2 then you win $15. If it lands on any other number, you lose $25.
- Homework 2 - Two teams decided to play football. They want to decide who receives first. Mike and Steve are the team captains. So, they have two suggestions to decide who receives the ball. Decide whether the suggestions are fair ways to make the decision.

### Practice Worksheets

Beware: There are a few loaded questions in here.

- Practice 1 - 3 friends are playing with dice. They decide to roll a die. If it lands on 1, then you win $30. If it lands on 2, then you win $35. If it lands on any other number, you lose $5. Is this game fair?
- Practice 2 - Two teams decided to play volleyball. They want to decide who serves first. So, they have two suggestions that who starts first. Decide whether the suggestions are fair ways.

### Math Skill Quizzes

I can't believe that people way in on these type of decisions on a daily basis.

- Quiz 1 - A local scout troop is having a cash raffle. Each ticket is $10. $2,000 is the grand prize. 2 prizes of $50 will be given out as well. You find out that 1,000 will be sold. How much will the average ticket holder lose?

### How to Use Probabilities to Make Fair Decisions?

The measurement of the possibility of an event taking place is what we call probability. One area where probability is very extensively used is the weather forecast. A smaller probability indicates that the chances of an event taking place are very low. The higher the probability, the greater are the chances of the event occurring. Probability is even used to make a fair decision, but how can we use this concept to make a fair decision? A fair decision is one where the possibility of all events happening is equal or if the expected value of a random variable is zero.

A case of fair decision is when a football coach must choose a Captain from the team under the right conditions. The situation of fairness will be if he considers everyone on the team for the position; this way the probability of each team member to become captain is equal. If he is choosing based on certain criteria that are not math based, this is an unfair decision.

Just because a decision is made unfairly, does not make it the wrong decision to make. Think back to the football coachâ€™s choice of Captain; would it make sense for him to choose a Captain that does not take the sport or his team seriously? Probably not! He wants a dedicated person to serve that role who will do a good job. This is where we need to determine if the math fits the situation. There are times when choosing a random leader make sense for the sake of time.

Mathematically an outcome to an event can be considered fair if the chance of it occurring is equal to the simple probability of the situation. Meaning that you have to divide the chances of an event happening by the total number of outcomes to calculate it the probability for all events is the same or not.