Although students have had informal experiences with probability in elementary school, their first formal introduction to the study of probability begins in seventh grade. Students conducted experiments or simulations and learned about calculating empirical (experimental) probabilities from their results. They made comparisons of empirical and theoretical probabilities and used theoretical probabilities to predict relative frequencies for a simple chance event. Students learned to set up probability models from a theoretical perspective (such as rolling two dice) or modeling a situation with or without technology and using the empirical data as an estimate of the theoretical outcomes. With compound events, students used tree diagrams, organized lists, or tables to generate the sample space for the event. They use the sample space to calculate the probability of a particular event occurring.
At the high school level, students will extend their knowledge of probability and use their previous experience with simulations as a basis for more complicated ideas. Students will use their experience in making sample spaces in middle school and extend the use of these sample spaces to determine the probability of an event occurring to including the probability of event A and event B occurring, of event A or event B occurring; and the probability of an event not occurring. Students will study a variety of problems in order to develop an understanding of independent and dependent events. They learn that the second event has not been influenced by what occurred in the first event when the two events are independent. The idea of conditional probability, i.e. finding the probability of one event occurring given that another event has already occurred, is studied in this unit and can be used as another way to view independence. In terms of probability notation, conditional probability can be expressed as P (A/B) = P (A and B)/P (B). In determining independence of events A and B, when P (A/B) = P (A) and P (B/A) = P (B), then A and B are independent. Another way to determine independence is through the construction of a two-way frequency table, used when two categories are associated with each object being classified. After all of these calculations, it is important to be able to recognize and understand conditional probability and independence and be able to explain theses concepts in everyday language.
Following exploring and understanding the ideas of conditional probability and independence, students are introduced to the rules of probability. These include the Addition Rule P (A or B) = P (A) + P (B) – P (A and B), and the Multiplication Rule P (A and B) = P (A) P (B/A). They should be able to use and interpret these rules. Additionally the use of permutations and combinations will help in determining probabilities and solving problems. Use of Pascal’s Triangle is particularly helpful in determining combinations.
When using applications of probability to solve problems, often a numerical quantity is more useful than a description of possible outcomes. Graphing a probability distribution gives a different perspective. This is where the expected value is introduced and interpreted as the mean of the probability distribution. Students will develop a probability distribution from a sample space in which the probabilities were assigned either theoretically or empirically. They will then calculate the expected value.
The use of expected values and probabilities are widely used to solve problems and evaluate decisions. Probabilities can be used to weigh decisions depending on the probability of outcomes. For example how much a company charges for an extended warranty depends on the cost of repairing or replacing an item and the probability that that item will fail. Games of chance depend on the ideas of expected value. Analyzing decisions and strategies involve probability concepts. Students should recognize the wide impact that probability can have on decisions they make.
