Unit 6: Probability Essentials

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 Teaching Tips

• We have found ourselves tempted in our own teaching practice to spend too much time teaching combinatorics, set theory, and "balls from urns" problems because that's the way we were trained and because we enjoy it. However, we want to warn you away from this practice should you be so tempted because probability, in this course, is solely used to study inference.

• Stress to your students that probability is a "long-run" frequency. For example, it would be meaningless to ask, "What is the probability that Iraq will be a 'western-style' democracy?"

• As elsewhere, don't focus on formulas, but on understanding. This could be illustrated perhaps by showing how futile it is to use the formula to find a conditional probability compared to thinking of the connection as a restriction of the sample space. A few simple contextual examples will help relieve students’ anxiety about independent events and prevent them from thinking of this idea as the application of a formula.

• If the question you are asking can't be done with a two-factor table or a tree diagram, then it is too complicated a problem for this course. Your students must be comfortable solving problems with two-factor tables and tree diagrams, but do not need to know counting rules or combinatorics. ("Factor" is another word for "categorical variable".)

• Some books define independence in terms of a product: Two events are independent if and only if P(A and B) = P(A) P(B). However, we feel you'll develop stronger intuitive understanding of independence if you use the definition in terms of conditional probability: two events are independent if the probability of A happening, given that B has happened, is the same as the probability of A happening. In other words, knowledge that B has or has not occurred has no effect on the probability A will happen.

• There is a certain class of problems (e.g. the Monty Hall problem) designed to exploit the fact that our intuition about probabilities is often wrong. If you are secure in your understanding of probability, then these problems are quite a bit of fun. However, for all others, these problems may send the take-home message that probability is hopelessly complicated.

• Language is as important in this unit as in others even though Probability is more math-oriented. The inclusion of a single word can drastically alter the problem. For example, "those who have stereos and TVs" is different from "those with TVs who have stereos." Students will need practice interpreting these type of statements. Always make explicit the differences between colloquial use of words like “OR” or “independent” and the statistical definitions.

• It is often useful to find real data that can be used to formulate probability problems to students. One can find two-by-two tables with categorical variables that will likely be of interest to students and formulate elementary problems for students to solve that will likely increase their understanding of the concepts more than traditional dice or balls in urns problems.

Student Misconceptions and Confusions

• Be careful to distinguish between mutually exclusive (or disjoint) events and independent events. Students often confuse the two, but they are very different ideas. The every-day use of the word "independent" is synonymous with "separate." But in Statistics the concept of "separate" is captured in "mutually exclusive." Statistically speaking, "independent" means "no relation between" or, more specifically, it means that knowing the outcome of one event does not affect the probability that the other occurs.

• Students will be confused by sentences with the following words or phrases: “or” and “and”, “with” and “at least” and “at most”