High School: Statistics and Probability

Using Probability to Make Decisions HSS-MD.B.6

6. Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

Mirror, Mirror, on the wall,

Which is the fairest of them all?

One of the best uses for probability is to help us understand whether or not events are truly random. This way, we can ensure that there is no bias on the part of any individuals involved, no bias on the part of the device being used, and therefore a "clean" and "fair" decision is being made.

Students should know that we use dice, coins, random number generators, lots, and many other things in order to ensure fairness. Still, there will always be deviation (and those people who claim they can roll double sixes with two fair dice ten times in a row).

Students should understand the factors that make decisions fair and random and be able to identify when such decision-making methods are biased or skewed. Students should also know that fairness and randomness are not necessarily synonymous, depending on specific situations.

In the classroom these questions can be demonstrated, simulated, and answered with many different examples. Here are a few ideas you can do to demonstrate them. First, come up with an example or a decision that needs to be made, then propose ways by which it can be made.

Are the decisions fair from a probability standpoint? Let the students decide after numerous tries to ensure randomness. Are the decisions fair from a moral standpoint (for instance, randomly deciding who fails the class based on a coin toss)?

• Toss a fair die
• Drop a fair die
• Toss an unfair die (but don't tell them about it)
• Select a randomly chosen card from a "fresh" deck of 52
• Flip a coin
• Use lots or straws
• Use a spinning top
• Use a random number generator

Discuss the strengths and weaknesses, randomness or bias, fairness or unfairness of each method from a mathematical standpoint. Discuss methods to undermine or enhance the fairness of each method, and when such methods will be useful in making decisions.

Drills

1. Which of the following decision-making method is fair?

Tossing a fair coin and using a random number generator

All other answers are clearly biased in one way or another. The slightly tilted roulette wheel would clearly land on the downward side more often, dropping a fair die is still unfair because of the way it's tossed, and papers of different sizes introduce bias into the decision making process. Only a random number generator would not.

2. Your friend wants to show you a magic trick and takes out a coin from her wallet. The coin is flipped several times in a row and all the results are heads. What do you think?

All of the above

There is no mention of fairness in the question. Therefore, the coin may be unfair or fair. Theoretically, a coin can flip the same way an infinite number of times and still be fair. It's statistically unlikely, but not impossible. She's either really good at manipulating the coin, it's unfair, or it's just a random result.

3. You and your entire class are stranded on a desert island. A rescue boat can save all of you except for one, who will be left on the island forever. You have all decided that whoever picks a number closest to the one that is randomly generated (luckily, someone brought a laptop) will stay behind. What should be done to ensure a fair result?

Make sure that everyone writes their guesses down

Allowing people to talk to one another can make it more difficult and allow people to conspire against one particular person. Looking at each other can either have no effect or can subtly convey messages to people via facial expressions, which could undermine the fairness. Writing down all the decisions ensures no foul play after the number generator reaches its result.

4. Lots that are used for a fair decision should be checked so that they do not have which?

All of the above

With different weights or sizes or markings, it wouldn't be a random lot. If lots are not random, they are biased and unfair. In this case, randomness translates to fairness. That means (D) is the right answer.

5. If a lottery picks a number that seems to come up too often, using the vacuum ball generator you may have seen on TV, then the possibilities for this are:

All of the above

The best way to determine if these occurrences are statistical anomalies is to investigate the ball in question. Until this is done, there is no way to know whether the ball is somehow different than the rest or if its selection is due to chance.

6. You and a friend are in a disagreement over who gets the last piece of pizza. What is the best way to fairly decide who gets to eat it?

Let a third party who doesn't care about you, your friend, or the pizza flip a coin

The best way is (C) because this way, we ensure that neither you nor your friend will try to manipulate the results by using an unfair coin or some other trickery. A third party may be biased if they make the decision themselves (maybe they like your friend's shirt more, which influences their decision), so which is the fairest of them all? Naturally, (C).

7. Why is a normal spinning top a fair way to make decisions?

It has no bias as to how it lands

A spinning top may have no distinct sides at all (if it's completely circular), so (A) can't always be true. On the other hand, assuming it's a normal spinning top, it should have no bias as to where it lands.

8. Is playing poker to decide an outcome of a dispute a fair way to do so from a statistical perspective?

No, psychology plays as much a role as the mathematics of the cards

In poker, while cards are randomly dealt, and your statistical advantage over time should be the same as your opponent's, psychological manipulation and experience plays a great role over results. As such, it isn't as fair as simply flipping a fair coin to decide an outcome, which would be a truly unbiased, random result.

9. You and 3 friends are driving out to watch a movie. You all want to watch something completely different. What is the fairest way to decide which movie all four of you will see?

Draw lots

The best way is to draw lots with one movie per lot listed. Flipping a coin is inappropriate as it will take more than one flip to decide an outcome in four-person ordeal, and rolling a six-sided die may also take more than one try since two sides will be unaccounted for.

10. When would a truly fair outcome statistically speaking be most unfair in reality?

When the outcome has serious consequences