Events, Sample Spaces and Probability
Finding the sample space is the key for student success in computing probabilities. Frequently problems arise when possible outcomes are missed, or double counted, or otherwise. That is to say, there are lots of less than intuitive possibilities so tools to support students finding correct answers are helpful. One of those tools is listed in the text, where a table is made with the different outcomes. Lists are also good, especially for sample spaces that only involve one item, like a single die. There is also no shame in drawing pictures or working with manipulative. I frequently visualize a deck of cards in my mind when working with those probabilities, but as a bridge player I’m comfortable with cards, while a student might not be. Don’t hesitate to give students cards to work with.
I encourage my students to never work in percentages. I realize that it’s kind of picky, but I find it useful to drive home that probabilities must be between zero and one inclusive. The strong restriction is very useful as it frequently provides feedback for mistakes. If you are computing any probability and it ends up outside of that interval, then something is wrong. It’s also a great tool to use to eliminate answers on a multiple choice test. I like to have a mantra to really drive it home. Between and and all possibilities add to .
Elementary set operations are a fundamental part of mathematics, but one that is taught in inconsistent places. I’ve always marveled at the fact that nearly all of my college textbooks, including my graduate level texts, start with a preliminary chapter on set theory. This has always led me to believe that set theory is either not taught, or is given an incomplete treatment in a lot of classes. There really isn’t any reason to give a deep treatment to it here. Sometimes sticking too much to the strict notation is going to cause more problems than it’s worth, since the ideas here are intuitive and students have worked with them in venn diagrams and other problems in the past.
An important thing to remember is that set operations are binary operators. That is, even if there are more symbols, only two sets can be operated on at a time . Due to the fact that the combination of unions and intersections is not associative (that is to say: I always include parenthesis if there is more than one operator, even if they are all the same operation, just to avoid confusion. This is, and matrices, are some of the first structures students encounter that do not follow all the classic rules of real numbers that students are used to. It can be an opportunity to push a talented class, but it is really an extra topic that has limited utility to the ultimate goal for a stats class.
The Complement of an Event
The classic example of value here is the birthday problem: In a given room, what is the probability that at least a pair of people have the same birthday? This is a great problem, as it shows how working smarter using some principles of probability makes a seemingly tough problem easy. It also has a result that is fairly counter intuitive; the probability of at least a match is much higher than one would presume. Both are key ideas to drive across to students studying probability for the first time. Asking for “at least one pair” means that if you were to directly calculate the probability it would take a very, very long time. There are just too many ways to get a match once the number of people in the room exceeds 4 or 5. However, asking the question “what is the probability that no two people share the same birthday” is logically equivalent to the first question and much easier to compute. Subtracting this quantity from 1 (finding the complement) then yields the answer.
The key here might be in re-writing the question in terms of the complement. Students probably don’t see what the big deal is in calculating complements, and that is really simple. However, making the complement work for you requires seeing where to apply it, and then what exactly you are looking for. The key hints are when a question is asking for a probability where multiple situations are possible. In the birthday problem, asking for exactly a single pair should be calculated directly, but at least a pair dictates that the complement is easier. Students should practice identifying and re-writing questions to make it so that time isn’t wasted attempting monumental calculations.
Students are going to get tripped up with the order of conditional probabilities. The more intuitive way of thinking of conditional situations is “if then” as opposed to “given”. While I am nearly always in favor of understanding the concept and avoiding formulae, in this case the formula is great. Because the order is a little strange, and frequently is mixed up, this is one of the few formulae that I put on a poster and ask students to commit to memory. Application is easy once the spaces are put in their proper place.
Additive and Multiplicative Rules
This is probably student’s first exposure to the dreaded problem of double counting. In this case, in contrast to the previous section, I don’t emphasize the formula here. Finding values or probabilities that are double counted is a huge part of statistics, and one that most stats students have memories of sitting in a group, having problems with getting the correct answer, pulling hairs out, only to then have someone say “double counting!” For this reason, I really try hard to get my students to understand where double counting occurs and try to train them to always be aware of where the error is likely to occur.
Basic Counting Rules
Combinatorics are the foundation of lots of basic probability questions. Cards are a great way to do look at problems, especially with the recent attention paid to poker. The television broadcasts will show probabilities for winning in “real time”. This can be used to practice and find those same percentages, which are relatively simple to compute. The foundation of each probability is finding the different cards that will allow for a win, against the total number of possible cards left. This is a nice extension of finding general probabilities for each type of poker hand.
Whenever combinitorics and probability comes up gambling is not far behind. This can cause problems in some cases considering the ethics of teaching typical gambling games in a classroom to students who are not legally able to wager bets. A couple of thoughts on the subject, and a general defense. First, it may be useful to make the distinction that while games are being taught and talked about, at no point is gambling going to occur. This is similar to a health class where effects of drugs are being discussed, but clearly no endorsement, or use, of drugs is happening. There is also historical context, as the earliest theories and work on probability was, in fact, motivated by gambling. The legacy remains, even in situations where gambling is no longer associated. For example, a hand with no high card points in bridge is called a Yarborough, named in honor of a lord who would offer his opponents a payout if no points were dealt (odds of getting a yarborough are , so the Earl made quite a profit on this). Second, I would promote the idea that gambling institutions profit from a lack of knowledge of probability. Like the Earl of Yarborough, the idea of casinos is to present a situation that looks favorable to the gambler where in actuality the advantage is firmly in the direction of the house. Knowing exactly how much of an advantage is disheartening to a gambler, and in many cases will cause a loss of interest in gaming.
While it is useful to compute a couple of combinations and permutations by hand to get a sense for how they work, I quickly pull out the calculators. In fact permutations and combinations might be the functions I use most often on the calculator right behind the trig functions. There is no benefit to spending any extra time with the tedium of not using the calculator functions.