Unit 6: Probability Essentials

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Connections

Probability
  • 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.
Inference
  • 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.