In this chapter we continue our discussion of probability by looking at probability models for random variables. A random variable is a numerical measure of the outcome of a probability experiment. Therefore, rather than looking at the probability of some experiment, such as heads or tails, we consider the probability of observing a specific number of heads. The chapter begins with a presentation of probability models presented in graphical and tabular form. Then, in Sections 6.2 and 6.3, we look at probability models in formula form (the binomial probability distribution and Poisson probability distribution).
What to Emphasize
Continue to emphasize the interpretation of all probabilities. Be sure you emphasize the probabilities as a foreshadowing of P-values. That is, emphasize the assumption going into the interpretation. For example, suppose we conduct a binomial experiment with n = 10 trials and probability of success p = 0.6. The probability of obtaining x = 7 successes is 0.2150. So, if we conducted this experiment 100 times and if the probability of success is 0.6 (the null hypothesis), we would expect to observe exactly 7 successes in about 21 or 22 of the repetitions of the experiment.
Now that we are getting into formulas for obtaining probabilities, you must decide what approach to take in obtaining the probabilities. That is, do you obtain binomial probabilities using the formula by hand, using tables, or using technology? We recommend using technology to find the binomial probabilities. It is certainly recommended that a couple simple examples be presented that require use of the formula. It is also recommended that the students understand the requirements of a binomial experiment along with the origins of the binomial formula. However, the emphasis of the chapter should be on model requirements and interpretation – again, as a method to foreshadow hypothesis testing. So, ask students for interpretation and to identify unusual results.
- Discrete Random Variables – This section provides an introduction to random variables along with a presentation of discrete probability distributions from a tabular and graphical point of view.
- I begin by distinguishing discrete and continuous random variables. Students might like the visual that discrete random variables can be represented as points on a number line, while continuous random variables (which typically result from measurement) can be represented as a continuum on the number line.
- Probabilities for discrete random variables are presented from both a tabular and graphical point of view. In addition, we discuss the mean (or expected value) and standard deviation of a discrete probability distribution. Be sure to emphasize the interpretation of the mean of a discrete random variable as the expected value. It should also be emphasized that the mean is the mean outcome of the experiment if the probability experiment is conducted many, many times (the Law of Large Numbers). It might be a good idea to simulate the results of a probability experiment with discrete outcomes. Plot the mean value of the random variable against the number of trials. o When discussing expected value of a discrete random variable (especially for the life insurance scenario), be sure to emphasize the fact that the expected value is not the outcome of a particular experiment. Rather, it is a long-term average outcome. For this reason, the insurance idea only works over many, many trials. For example, if a company only insured fifty individuals, the company would go bankrupt with a single claim.
- Finally, note the technique used in graphing probability distributions. The approach utilized is meant to emphasize the fact that discrete random variables have specific outcomes and the distribution is not continuous.
Expected Value The following table shows the profit from a $1 bet in a video poker game.
|Four of a Kind||0.00225||$39|
|Three of a Kind||0.06883||$2|
|Jacks or Better||0.18326||$0|
|Less than Jacks or Better||0.58076||-$1|
Source: Barnett, Tricia, “How Much to Bet on Video Poker,” Chance, Vol. 24, No. 2, Spring, 2011
(a) Determine the expected value of the game.
(b) Suppose a player can play 30 games in one hour. What are the expected losses of the player after 6 hours of playing?
- The Binomial Probability Distribution – This section presents the binomial probability distribution. We consider this an important section as a preview of inference on proportions.
- Be sure students verify that the requirements for a binomial experiment are satisfied for each probability experiment.
- As mentioned earlier, I recommend spending time on explaining the origins of the binomial probability distribution function. It is also worthwhile to use the formula for some simple probability experiments. However, we also recommend that technology be used to obtain binomial probabilities for a majority of the problems. This will allow for time to be dedicated to explaining what the results represent. Developing this skill will serve the students well when it comes time to look at hypothesis testing from the P-value point of view.
- I also recommend that binomial probabilities be computed based on results of a poll. For example, suppose nationwide that 36% of adult Americans have nothing saved for retirement. Further suppose you conduct a survey of 100 Americans with a bachelor’s degree and find that 45 have nothing saved for retirement. Use the binomial probability distribution function to determine the likelihood (probability) of obtaining 45 or more successes in 100 trials of an experiment in which the probability of success is 0.36. Is the result unusual? What might we conclude based on this result? Another area of discussion: notice that a “success” in this experiment is not something positive (here, a success is finding an individual who has nothing saved for retirement).
- Finally, spend time discussing the roles that n and p play in the shape of the distribution of a binomial random variable. Consider using the Binomial Distribution applet in StatCrunch for this.
A Drug Study Toxic Epidermal Necrolysis (TEN), also known as Lyell’s syndrome, is a life-threatening disease characterized by blisters that cover the skin and extensive peeling off of skin. In a double-blind, randomized, placebo-controlled study of the effects of a drug called thalidomide on TEN, it was found that 10 of the twelve patients in the thalidomide group died compared with three of ten in the placebo group. Using the methods of this section, explain why the study was stopped. What does this result suggest? Source: Randomized Comparison of Thalidomide Versus Placebo in Toxic Epidermal Necrolysis, Wolkenstein, Pierre et al. The Lancet, Volume 352, Issue 9140, 1586-1589. Answer: P(X > 10) = 0.0002 assuming p = 0.3.
- The Poisson Probability Distribution – This distribution may be skipped without loss of continuity. I do not cover this material in my classes.
- As with binomial probabilities, we recommend using technology to find probabilities. o Continue to emphasize the interpretation of probabilities in this section and use the probability distribution function to identify unusual results as a way to foreshadow P-values.