I. Curricular Framework

Concept:

Prediction

Essential Question:

What occupations rely on the principles of statistical probability?

Bridge:

Observations

Content:

Statistical Probability

Outcomes:

II. Standards Aligned

III. Instruction and Assessment

1. Connect: Connecting to the Concept Experientially

Objective: To provide students with a personal experience of chance.

Activity: Students test their own extra-sensory perception. Students use index cards to make decks of test cards using the standard testing symbols for ESP: square, circle, star, cross, wavy lines. Each deck should contain twenty-five cards, five of each symbol. In groups of three, one student draws a card from the deck, and concentrates on the symbol. One student draws a card from the deck, and concentrates on the symbol. The second student, who is the test subject for this round, tries to receive the mental image from the first student. The third student records both the drawn symbol as well as the "guess" of the subject. Have the students try this experience in several ways:
1) shuffling the deck, then going through the whole deck one card at a time.
2) shuffling the deck, then drawing one card at a time, replacing it and reshuffling for each trial.
3) shuffling the deck, then randomly drawing one card at a time from the deck, but not replacing it or reshuffling for each trial. Let them repeat the test each way for twenty-five draws. Then the students in each group switch roles until each group member has been a transmitter, a receiver and a recorder. Allow groups to discuss their results.

Assessment: Participation, helpfulness and interest.

2. Attend: Attending to the Connection

Objective: To enable the students to examine the process of testing their ESP abilities.

Activity: Have each group select a reporter who will explain to the class what his/her group discovered about their ESP. Reporters and class members can respond to the following discussion questions: How well did you do? Does anyone in your group have ESP? How are you able to tell? Did anyone get a perfect score (all correct)? Do you think it takes a perfect score to indicate ESP? What would be a good score (likely to have ESP)? Teacher should then mention the concept of comparing desirable results (correct guesses) to the total number of guesses, using a ratio to compare these numbers.

Assessment: Quality of discussion.

Assessment, Phase One, Level of Engagement, Fascination:

3. Image: Creating a Mental Picture

Objective: To have the students experience the inaccuracy of their guesses as to the likelihood or unlikelihood of certain events.

Activity: Ask the students to guess the probability of at least two students in the class sharing the same birthday. It would appear that this is a very unlikely event. However, in a normal sized class of twenty-five students, it is almost certain (nearly 100% probability) that there will be a pair who share a birthday. Ask each student to call off his/her birthday. If another student hears his/her birthday called, s/he should raise her/his hand. Of 365 days which are possible birthdates, we are considering pairs of birthdates. With one person, there are no pairs. With two persons, there is one pair. With three persons, there are three pairs. For each additional person in the group, another pair is added for each person already in the group. Therefore, to determine the number of pairs for a group of n members, you must add the sum of all numbers from 1 through n-1. As you can see, there are certainly more opportunities for a matching pair than appeared at first. Now try another activity. Ask students to jot down a number between 1 and 50. If you have a group of up to 15 students there will be a 90% probability that two students will select a pair of matching numbers. Students will usually assume that it is fairly unlikely for a pair of students to select the same number. Have students call out their numbers as with the birthdays to see if any pairs exist. For larger classes, extend the range. Now ask the students what these two examples demonstrate about probability. There is likely to be a variety of responses. If no one makes the point that we must have some standard procedures to determine the probability of certain events, the teacher should stress this, and tell the students that the purpose of this unit is to learn about these concepts and procedures. Probability is a ratio comparing the frequency of events to the total number of events. The underlying problem in most cases is to determine these numbers.

Assessment: Quality of discussion and accuracy of understanding.

Assessment, Phase Two, Seeing the Big Picture:

4. Inform: Receiving Facts & Knowledge

Objective: To make students aware of the basic concepts of probability.

Activity: Teacher lectures on the following topics: definition of probability, sample space, equally likely events, complementary events, odds, independent events, dependent events, event, theoretical probability, experimental probability, textbook readings and assignments.

Assessment: Teacher checking for understanding; student attention and ability to solve in-class problems.

Assessment, Phase Three, Success with Acquiring Knowledge:

5. Practice: Developing Skills

Objective: To give the students additional practice in solving probability problems.

Activity: Workbooks, worksheets and textbook problems.

Assessment: Quality and accuracy of the above. Objective test.

Assessment, Phase Four, Success with Acquiring Skills:

6. Extend: Extending Learning to the Outside World

Objective: To allow the students to try several hands-on experiments and compare the calculated experimental probability to the theoretical probability.

Activity: The teacher sets up a series of experiment stations. Students are instructed to write a report on each experiment. The report must include a short description of what event is being examined in which the students should describe the favorable event in the experiment, figure out the theoretical probability of that favorable event, perform the experiment 20 times and record results, calculate the experimental probability based on results, write a short analysis of the results. Compare the theoretical and experimental probabilities. If there is a great discrepancy, how can it be accounted for? What other factors influenced the results? The experiments for each station are as follows: Rolling a single die to come up with "5." Drawing a certain colored marble from a jar containing a mixture of several marbles in various colors. Flipping a coin to come up "heads." Drawing a face card from a deck of 52 playing cards.

Assessment: Quality of the reports.

7. Refine: Refining the Extension

Objective: To have students select a particular application of probability theory, and to investigate this application.

Activity: Give the students the following four options of which they must choose one and write a report: Research the uses of probability in gambling. How do casinos know that they are going to make money? How do casinos set the odds? Go to the library for information on this subject, or write to Nevada casino for information. Talk to a local church or civic group that runs Bingo or other charitable gambling events. Survey your classmates about some of their genetic traits. Sample the students in this class about various traits which they inherited genetically. Some possible traits to consider include: left/right handedness, connected or nonconnected ear lobes, ability to roll the tongue, color blindness, blood types, hair or eye color. Using your results project the probability that such traits will occur in the general population. See if you can determine some actual statistics from the research section of the library. If your results are different from the listed statistics, state why you think this occurred. Study the use of probability in the insurance business. Conduct library research and talk to an insurance agent, an actuary, or someone who is involved in setting up insurance policies, so you can get first hand information on how probability and statistics are involved in these fields. Open your eyes and ears and locate some other application for probability theory in business or science. You must have your topic approved by the teacher.

Assessment: The level of the students' involvement.

8. Perform: Creative Manifestation of Material Learned

Objective: To have students present their project.

Activity: Students present their project to the large group


Assessment: The level of the students' relation of the project to the concept.

Assessment, Phase Five,Performance, Creative Use of Material Learned: