Life is stochastic, and those who refuse to embrace uncertainty will be doomed to fear it. Insurance companies, successful investors, and competent leaders use their understanding of statistics and probability to make decisions that result in the best expected outcome possible. Ok, there are no competent leaders, but then that's the point of this article, isn't it?
If you are teaching mathematics, you are putting tools in the toolboxes for tomorrow's professionals. The problem is that math teachers consistently fail to convey the idea of concepts as tools. Students feel they are being asked to play a mind a game and quickly lose interest, only studying enough to get by on the exam.
In courses like college algebra, it is virtually impossible to get students interested. Of course, I use my understanding of functions every day in my trading, but even if I had wanted to share my options strategies when I taught, I certainly didn't have time to give lectures on finance.
Statistics and probability are different. At a basic level, the ideas are so concrete that you can easily catch the students' attention. Here are a few ways to do that:
1. Have students survey each other. Each student team is a marketing agency with a product and an advertising budget. Do they want to buy web ads? Advertise on TV? Which sites? Which shows? They will need to determine the statistical likelihood that an individual who might buy their product might watch a certain show or go to a certain web page.
By the way, this is a great way to contrast statistics with probability. You cannot count all potential consumers and quantify their behavior as you would with a deck of cards or a bag of jellybeans. The numbers you get from a survey are imperfect measurements, and hence are not true probabilities.
2. Use salaries or household income to contrast mean and median. Do an example where a hypothetical rich person moves into a poor neighborhood. What happens to the median income? The average? Which is a better estimate of the socioeconomic well-being of that neighborhood? Which would you quote if you were trying to argue for government assistance in that area? If you were trying to argue against government assistance in that area?
3. Have the students count a sample of objects, such as M&M's. Pool the results and discuss variation in the samples. If you give each student a handful of M&M's, the ratios of the colors should be fairly consistent, but there will be enough variation to make a good point about sampling and error.
4. Use decision squares to make decisions. A decision square is has four cells in the simplest case, and is used to decide between two courses of action when two different natural outcomes are possible. In finance, binary options can be analyzed this way, although the vast majority of options are not binary, so it is of limited use. A good example would be the decision of whether to purchase a warranty plan.
Say your new stereo has a 5% chance of being defective. The warranty plan costs $20, and repairing the defective stereo would cost $60. If you pay for the warranty and the stereo is not defective, you lost $20 on the warranty and got nothing for it. That has a 95% chance of happening. You have a 5% chance of using the warranty, but either way, you will have paid twenty dollars. Hence, your expected loss if you buy the plan is $20.
If you don't buy the plan, you have a 95% chance of paying nothing for repairs, and a 5% chance of paying $60. Your expected loss if you don't buy the plan is thus (.95)(0)+(.05)(60), or $12.
From a realist's point of view, the warranty is a bad deal. A pessimist who expects the worst to happen no matter what will buy the warranty, while the optimist will shoot for the best possible outcome. A pessimist looks for the most acceptable loss, while an optimist looks for the largest possible gain (or best possible outcome). Neither has the wisest outlook.