# High School: Statistics and Probability

### Using Probability to Make Decisions HSS-MD.B.7

7. Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Students should understand that when they're being asked to understand statistics, we're expecting them to understand statistics. Textbook mathematics does not equate to real-world problem solving abilities. Knowledge does not imply understanding.

It's easy to jot down a formula here and there (or punch it into a TI-89) and plug in the x and y to get z. It's a lot more challenging to understand what x, y, and z are, and why we use that formula and not another one.

Whether students know an equation or not is rarely relevant. Equations can always be looked up. What's more important is that students are able to logically analyze a situation (without any math as part of a problem), and explain why a certain approach should or shouldn't be used.

This forces students to not only analyze the problem itself, but also to justify their approach from a logical perspective. Logic, in the world of statistics, can only come about from a deep knowledge of fundamental concepts applied over and over again to different real world scenarios. Only then does the answer become "logical" as opposed to "difficult."

Students should eventually reach a point where instead of struggling with every problem, they react with, "Duh! Obviously!" and quickly move on. This is a sign that they've learned and internalized the logic behind the problems. Or that they're cheating.

That's why it's not enough to just say, "Duh! Obviously!" and scribble down an answer. Students should be able to explain why their answer is not only correct, but logical as well.

Some examples for classroom exercises and learning approaches include:

• Critiquing decision-making process before, during, and after an event's occurrence based on simple probability concepts. This can include decisions in sports, research, and so on.
• A class-wide discussion on a particularly difficult approach to analyzing a situation (such as a science article). Look into how they accounted for confounding factors, what the confounding factors could have done to the probabilities involved, and so on. Do the results actually provide any insight into the supposed population? Was the sample correctly identified and gathered? How would this influence results and probabilities?
• A critique of an article with a poll in it. Find out the source and methodologies of the poll to understand why it may have been incorrectly conceived and how the results may not actually represent the journalist's supposed population involved.

#### Drills

1. A tire manufacturer knows that out of every 100,000 tires, it will have 1 blowout on the highway. This will cost the company about \$1,000,000 in claims damages per blown out tire. Is it a good strategy for the tire manufacturer to charge \$10 per tire in order to cover the litigation?

No, because the tire company has expenses other than litigation

The probability in the problem is empirical, but it could still be calculated incorrectly and is not immune to statistical irregularities. While the price of \$10 per tire covers the cost of the \$1,000,000 litigation, (C) is right because the company has expenses other than litigation. Basing the price of the tires off the cost of litigation alone would be silly.

2. A research company is testing four different drugs (the legal kind). The people chosen are randomly selected for the study but are not randomly assigned to the different classes of drugs. Despite randomly being selected for the research, why is not assigning drugs at random still dangerous?

This may introduce selection bias based on the researcher's preferences

While (C) is true, the question asks why not doing so is still dangerous. The lack of random assignment introduces bias into the study, so (B) is true. However, in cases such as these, it is more important to care for the patients' health than make sure the study is bias-free. When it is impossible to remove all biased or confounding factors, these must be accounted for statistically when the results are obtained.

3. Market research organizations usually develop testing guidelines for their products where they survey a sample of the population to figure out whether or not the product will be popular. Which of the following is best to do?

All of the above

This really depends on what the company is trying to do. Is it trying to appeal to its fan-base and find out the probability of hardcore supporters of buying this new product? To the whole population for the same reason? Or to wipe out the competition? Depending on the purpose, sampling a select group or a random set may prove the most useful to the company's study.

4. A clothing firm has just come up with a brand new style of jeans that they think will appeal to 25% of the population. The executives have decided that the chances that the item will be bought by a random person or not be bought by a random person are 50%, which they think is a great sales rate and will make them even richer. Are they correct or incorrect in their calculation?

The chances are actually 100%

Notice the wording: "the chances that the item will be bought by a random person or not be bought by a random person are 50%." Customers will either buy the jeans or not. The chance of one or the other happening is 100%, not 50%. Subtle, but tricky.

5. An oil company has hired a geologist to conduct a survey of newly purchased land. The geologist has never been to this part of the world and is unsure, given all the geological conditions, what the probability is of finding viable oil. To increase his chances of finding oil, which of the following should he not do?

Use a random sample survey and drill wherever he can to collect data

In this case, the best thing he can do to increase the probability of finding oil is to first educate himself on the probability of finding it through consultations. Random sampling is rarely approved for use; that's why the oil company hired an expert and didn't conduct the random sampling themselves. It's so they don't have to waste money on completely random guesses that could cost time, money, and adversely impact the environment.

6. The chances you will be struck by lightning are about 1 in 500,000 (depending on who keeps track). Would it be fair to say that if you and a group of 500,000 people were out in a field in the middle of a thunderstorm, only one of you would get struck?

No

There are many factors at play here. Simply having a "herd" protection mentality isn't good enough based on statistics, since the numbers only describe one chance event. For example, if you were standing in a soaking field with 500,000 people, lightning would only have to strike one point in the field to electrocute everyone on it. (Remember, water conducts electricity.) This is an example of how to be careful interpreting statistics and how they were gathered, what their data actually imply, and what assumptions these statistics make.

7. A marine sniper has a faulty weapon and needs to stay alive in a combat zone. He is able to hit a target that is 1 mile away 50% of the time, and a target that is 100 feet away with only 10% accuracy (it really is a faulty weapon). Assuming 10 targets far away and 2 targets 100 feet away, which should he target first in order to stay alive?

Those closest to him, to avoid being shot

Even though the probability of hitting those closest to him is lower, they are the greatest threat to his life right now. This is an example of when probabilities should not overrule common sense.

8. If the expected value of one round of golf after placing a bet with a friend is \$50, but it costs \$55 to play, which of the following is true?

All of the above

It's a personal decision here. From a purely financial standpoint, (A) is correct, but there's more to life than numbers. Also, the expected value can only give you an expected value, not the true outcomes. That means (B) may also be true, since sometimes you might win. Any and all of these answers may be right.

9. How can a firm with a new product best come up with important sales probabilities?

Research and focus groups

The best way to come up with empirical probabilities is through random sampling of populations. This is done through research and focus groups. Economists might be intelligent and make educated guesses, but they cannot provide empirical data specific to the company.

10. In general, which of the following is true about making a decision based off of larger samples?