K-12 Math Curriculum — Where's the Probability Theory?by StarbuckStudents go through 12 years of schooling and don't learn enough probability theory to be able to evaluate the barrage of probabilistic claims that will be thrown at them for the rest of their life. Evaluating probabilities is a life skill, and it needs to be taught. What we call common sense is developed early in life. The notion that we have to be able to read and write is part of our common sense, given the extensive attention given to those skills in school. The notion that we should be able to analyze probability claims is nonexistent in our common sense, despite the fact that in the technological world we live in, probabilities are thrown at us as arguments on a regular basis and we frequently just choose to believe what we are told. Let's look at two probability problems. The Monty Hall ProblemThis example looks whimsical, but the situation did occur as one of the many games on the TV show Let's Make a Deal many years ago. This is a well-known problem. A game show host will allow you to open one of three doors. Behind one door is a new car; behind each of the other two, a goat. You pick a door but you don't open it, and the host then opens (randomly, if he has a choice) a different door which he knows hides a goat. He then says, "Now would you rather have the unopened door you did not pick?" Should you switch your choice? Call the door you originally picked "door A". You were informed in advance that the following are equally likely possibilities:
Remember, the door you picked was door A. For each of the cases (a), (b), (c), these are the results of choosing to switch:
So if you switch, 1/3 of the time you'll get a goat, and 2/3 of the time you'll get a car. Therefore your best bet is to switch. The crux of this problem is that the first thing you have to do is focus on what things are equally probable. You have to get that right before you start reasoning. People given this problem might say, staring at the two closed doors after the host has opened one door, that because there are two closed doors, they must (?) have equal probabilities of hiding a car; but there is no reason in the world to think that. What the host is really doing here is offering you the choice of sticking to your original door or of being able to open both the remaining doors; his opening one of them is irrelevant. (Are there variations on this problem? Yes — see http://en.wikipedia.org/wiki/Monty_Hall_problem and scroll down to the section entitled Variants — slightly modified problems.) The Cancer TestOn the basis of his family history, lifestyle, and medical history, a male patient is deemed to have a 2% likelihood of having X-cancer. Because he now has a small tumor, he is given an X-test, which unfortunately yields a positive result.
Question: what is the probability that this man has X-cancer? Suppose we have 1000 patients who, like this man, have a 2% likelihood of having X-cancer. They now all have a small tumor and are all given the X-test. So how many of them get a positive result on the X-test, and how can we use that information to answer the question? 2% of the 1000, 20 men, actually have X-cancer: 98% of the 1000, 980 men, do not have X-cancer: So 16 + 98 = 114 men get a positive X-test result, but only 16 of them actually have X-cancer. So on the basis of the patient's characteristics plus the X-test result, our patient has a probability of 16/114 = 14% of having X-cancer. The crux of the issue here is that when we want to know what some probability is, we don't just wave our hands and say it's equal to the probability of something entirely different. People given this problem might be tempted to say that a positive X-test result means that the patient has roughly an 80% chance of having X-cancer; but there would be no justification whatsoever for such a claim, and the correct probability in this particular case, 14%, is very different from that. We need to be able to work out the number of men who would get a correct positive result and the number who would get an incorrect positive result, and then from that we can calculate the required ratio in a way that makes sense. RemarksIf you're a mathematician, you might be saying that the above problems could be solved using the appropriate Bayesian formula for conditional probabilities. Yes, fine. But kids in school need to first handle problems with numbers of occurrences and to concentrate on what assumptions make sense and what assumptions do not. We learned how to combine fractions years ago and we've forgotten how hard that was — it was very hard, but we learned it. And kids in primary school can learn to do probability problems like the ones above. With some background work like that, secondary-school students can go on to learn to deal with the intricacies of a wide variety of probability problems. The required time would probably be on the order of eight weeks. You don't think that all this can be done over the course of 12 years in school? Well then, guess what — it'll never get done. And newspapers, magazines, medical personnel, and financial planners will continue to swamp us all with probability claims that we have no idea how to evaluate. For more articles, see Want to learn how to play & win at chess, sudoku, poker, crossword puzzles, scrabble, checkers, or bridge? See
Last updated March 26, 2011
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