- This unit is the first comprehensive unit that discusses
probability. These concepts, specifically independence and conditional
probabilities, will reappear in several future units.
- Be sure to point out to students that the definition of
probability is the number of favorable outcomes of the event divided by
the total number of outcomes, which is similar to the definition of
relative frequency of an observation in a data set. They differ because
probability is a theoretical proportion for the entire set of possible
outcomes while relative frequencies are observed proportions for a data
set, which is frequently a subset of a larger set.
- The relation to statistical inference is that the
probability of an event is a theoretical model for the long term
behavior. For example, half the time one rolls a die it should yield an
even number. Then, one might collect data: perhaps one rolls a die 80
times and gets an even number 70 times. Statisticians try to set a
critical point at which we no longer believe the theory because of the
information provided by our sample data.
- Independence of events differs slightly from independence
of random variables, which students will learn in the next unit.
- As mentioned above, probability and inference go hand in
hand. This will be evident in future units. Specifically, conditional
probability is a key component of any hypothesis test.
- Many statistical tests are essentially tests about the
independence of two random variables. Two examples that you will study
later are the goodness of fit test and contingency tables. Presenting
students examples of these might aid in their understanding of
independence For example, answering the question “does living near a
power plant increase one’s risk of getting cancer?” would involve a
test of independence.