Probability

This chapter presents an introduction to probability.  We begin with a presentation of the various techniques that may be used to assign probabilities – the empirical approach, the classical approach, and the subjective approach.  In the next few sections, we discuss compound probabilities through the Addition Rule and Multiplication Rule.   We then present conditional probabilities and counting techniques.  We end the chapter with the first Putting It Together section.  This section represents a mixture of the various techniques used to assign probabilities to events. Students must first decide the type of problem represented and then use the appropriate technique for assigning the probability.

What to Emphasize

As you proceed through this chapter, the most important concept to emphasize is the interpretation of probabilities.  It is absolutely essential that you emphasize the interpretation of probabilities through the lens of relative frequency.  For example, if the probability of observing an event is found to be 0.45, you should require the following interpretation: “If we conducted this experiment 100 times, we would expect to observe the event 45 times.”   This is important as it foreshadows the interpretation of P-values when we get to hypothesis testing.

Second, there is much discussion in the statistics education community regarding the amount of emphasis that should be placed on probability.  The American Statistical Association recommends that probability be de-emphasized.  In other words, only cover the probability that is needed for inference.  For example, if you plan to cover the Binomial Probability Distribution, you will need to cover Sections 5.1, 5.2 (Addition Rule for Disjoint Events and Complement Rule only), and Section 5.3.  The organization of the text is set up to allow flexibility in various philosophies regarding probability.

  • Probability Rules – This section introduces three techniques for assigning probabilities (Empirical, Classical, and Subjective).
    • I begin with empirical probabilities as it provides an easy transition of interpreting probabilities from a relative frequency point of view.  In addition, it allows for an easy discussion of the Law of Large Numbers. Begin the section by asking a simple question, such as, “If I flip a coin, what is the probability of obtaining a head?”  You will likely hear answers such as “one-half” or “fifty-fifty.”  Ask students to explain how they obtained this result.  When a student replies that there are two outcomes and only one is a head, you have the opportunity to explain the idea of equally likely outcomes.  Prepare an applet in StatCrunch that has a weighted coin.  Make the probability of heads equal to 0.25.  Flip the coin 1000 times (1000 runs).  Highlight Convergence so students can see the relative frequency of head approach 0.25.  Start a discussion about the dangers of assuming equally likely outcomes.
    • Emphasize interpreting probabilities from a relative frequency point of view.

Quote by Pierre-Simon Laplace:   “To discover the best treatment to use in curing a disease, it is sufficient to test each treatment on the same number of patients, while keeping all circumstances perfectly similar.  The superiority of the most beneficial treatment will become more and more evident as this number is increased, and the calculus will yield the corresponding probability of its benefit and of the ratio by which it is greater than the others” Laplace, P.-S. (1825) Essay on Probabilities , 5th edn. Paris: Bachelier

(a)    What does Laplace mean when he says “while keeping all circumstances perfectly similar.”

(b)    Explain the meaning of “The superiority of the most beneficial treatment will become more and more evident as this number is increased,…”.  What law does this statement utilize?

  • The Addition Rule and Complements – Now we begin a discussion of compound events involving the word “or.”  Be sure to emphasize that probabilities involving the word “or” are assigned using the Addition Rule.
    • I only cover the Addition Rule for Disjoint (Mutually Exclusive) Events and the Complement Rule because the General Addition Rule is not needed for inference or the Binomial Probability Distribution.
    • Continue to emphasize the interpretation of probabilities from a relative frequency point of view.

 

  • Independence and the Multiplication Rule – These are probabilities involving the word “and.”  Emphasize the “and” probabilities utilize the Multiplication Rule.
    • I recommend coverage of the entire section – especially if you are planning to discuss the Binomial Probability Distribution.
    • Emphasize the idea that independence may be assumed provided the sample size is no more than 5% of the population size.   Most samples are from finite populations, yet the assumption of independence is made so that the mathematics is easier (and this assumption does not significantly affect the results).
    • Continue to emphasize the interpretation of probabilities from a relative frequency point of view.
    • I also like to emphasize that events that seem unusual on the surface (such as winning the lottery twice) are not that unusual when the population from which the event is drawn is large (there is a large population of existing lottery winners who continue to play, so it is not unusual to find a person who wins the lottery more than once).

Chevalier de Méré The following is a famous problem in probability. Antoine Gombaud (1607-1684), better known as Chevalier de Méré, who was a prolific gambler in the 17th Century.

(a)  What is the probability of rolling a pair of sixes when rolling two die?

(b) What is the probability of not rolling a pair of sixes when rolling two die?

(c) What is the probability of not rolling a pair of sixes in two consecutive tosses?

(d) What is the probability of not rolling a pair of sixes in twenty consecutive tosses?

(e) Use technology to determine the number of throws of a pair of dice for the probability of rolling a pair of sixes to exceed ½.

 Earn More Than Your Parents?  In 1970, 92% of American 30-year-olds earned more than their parents did at age 30.  In 2014, only 51% of American 30-year-olds earned more than their parents did at age 30.  Source:  Wall Street Journal,  December 8, 2016

(A) What is the probability a randomly selected 30-year-old in 1970 earned more than their parents at age 30?

(B) What is the probability that two randomly selected 30-year-olds in 1970 earned more than their parents at age 30?

(C)  What is the probability that out of ten randomly selected 30-year-olds in 1970, at least one earned more than their parents at age 30?

(D) What is the probability that out of ten randomly selected 30-year-olds in 2014, at least one earned more than their parents at age 30?

  • Conditional Probability and the General Multiplication Rule – This represents a continuation of the material from Section 5.3.  This material is optional and may be skipped without loss of continuity.  I do not cover this material in my classes.
    • It is true that we present conditional distributions when discussing contingency tables, but the formal presentation of conditional probabilities is not needed for that section.

 

  • Counting Techniques – Here is a presentation of various counting techniques using permutations and combinations.  This material is optional and may be skipped without loss of continuity.
    • Although this material is optional, it may not be a bad idea to discuss combinations if you plan to present the Binomial Probability Distribution.  Plus, combinations may be used to quantify the number of simple random samples of size n that could be obtained from a population of size N.

 

  • Putting It All Together: Which Method Do I Use? This is a great section for students to practice determining the appropriate method to use in assigning probabilities.  Be careful in selecting problems if you skipped any sections or objectives.