Probability Concepts and Applications
Teaching Suggestion 2.1: Concept of Probabilities Ranging From 0 to 1.
People often misuse probabilities by such statements as, “I’m 110% sure we’re going to win the big game.” The two basic rules of probability should be stressed.
Teaching Suggestion 2.2: Where Do Probabilities Come From?
Students need to understand where probabilities come from. Sometimes they are subjective and based on personal experiences. Other times they are objectively based on logical observations such as the roll of a die. Often, probabilities are derived from historical data—if we can assume the future will be about the ...view middle of the document...
Students often have problems understanding the concept of random variables. Instructors need to take this abstract idea and provide several examples to drive home the point. Table 2.4 has some useful examples of both discrete and continuous random variables.
Teaching Suggestion 2.7: Expected Value of a Probability Distribution.
A probability distribution is often described by its mean and variance. These important terms should be discussed with such practical examples as heights or weights of students. But students need to be reminded that even if most of the men in class (or the United States) have heights between 5 feet 6 inches and 6 feet 2 inches, there is still some small probability of outliers.
Teaching Suggestion 2.8: Bell-Shaped Curve.
Stress how important the normal distribution is to a large number of processes in our lives (for example, filling boxes of cereal with 32 ounces of cornflakes). Each normal distribution depends on the mean and standard deviation. Discuss Figures 2.8 and 2.9 to show how these relate to the shape and position of a normal distribution.
Teaching Suggestion 2.9: Three Symmetrical Areas Under the Normal Curve.
Figure 2.14 is very important, and students should be encouraged to truly comprehend the meanings of ±1, 2, and 3 standard deviation symmetrical areas. They should especially know that managers often speak of 95% and 99% confidence intervals, which roughly refer to ±2 and 3 standard deviation graphs. Clarify that 95% confidence is actually ±1.96 standard deviations, while ±3 standard deviations is actually a 99.7% spread.
Teaching Suggestion 2.10: Using the Normal Table to Answer Probability Questions.
The IQ example in Figure 2.10 is a particularly good way to treat the subject since everyone can relate to it. Students are typically curious about the chances of reaching certain scores. Go through at least a half-dozen examples until it’s clear that everyone can use Table 2.9. Students get especially confused answering questions such as P(X ( 85) since the standard normal table shows only right-hand-side (positive) Z values. The symmetry requires special care.
Alternative Example 2.1: In the past 30 days, Roger’s Rural Roundup has sold either 8, 9, 10, or 11 lottery tickets. It never sold fewer than 8 nor more than 11. Assuming that the past is similar to the future, here are the probabilities:
|Sales |No. Days |Probability |
| 8 |10 |0.333 |
| 9 |12 |0.400 |
|10 |6 |0.200 |
|11 |2 |0.067 |
|Total |30 |1.000 |
Alternative Example 2.2: Grades received for a course have a probability based on the professor’s grading pattern. Here are Professor Ernie Forman’s BA205 grades for the past five years.