The Whole Is The Sum Of Its Part
An Investigation of Reliability
Targeted Grades: 7 - 8
Master Teacher: Christine Bodner
OVERVIEW
How do engineers evaluate the reliability of a complex machine? In this lesson, the students will simulate and determine the probability of failure of an individual machine part, then learn how to use this information to predict the reliability of multiple parts working together. The Challenger Space Shuttle explosion serves as a real-life application of the mathematical concepts presented in this lesson. (Time: 45 - 60 minutes)
LEARNING OBJECTIVES
The students will be able to:
1. Explain the probability of failure as 1 - P (success).
2. Explain how to determine the probability of independent events.
3. Carry out a simulation to determine the probability of failure of both single and multiple machine parts.
4. Compare experimental and theoretical probabilities.
5. Make reliability predictions based on experimental or theoretical probabilities.
6. Describe a real-world application of quantitative probability.
MATERIALS
"Focus for Viewing" worksheets: one per student.
Dice: six standard dice for each working group (groups of three recommended).
Paper for calculations.
Calculators for multiplying probabilities (can be shared within groups).
VIDEO
Against All Odds: Inside Statistics: Random Variables (Program 16)
The Annenberg/CPB Collection.
PREREQUISITE KNOWLEDGE and SKILLS
1. Basic understanding of probability: definition, notation, range from 0 to 1; (0% to 100%).
2. Fraction and decimal multiplication.
3. Calculator skill in multiplying fractions and decimals.
PRE-VIEWING ACTIVITIES
(If you have access to closed-captioning on your TV, use it, if possible. It is an excellent learning tool.)
Cue video to approximately one minute into the program: you'll see an earth-colored frame showing " -3- Mean and Standard Deviation." And hear a woman's voice saying, "... can be used to describe random variables.")
Discuss reliability of familiar machines: ball-point pen, bicycle, computer, car, airplane. Students should exchange experiences and be urged to share their methods of predicting breakdowns. Ask students to discuss how a quantity, or numerical probability, assigned to the chance of failure of a pen, computer, airplane, etc., might affect the purchase or use of these machines.
Review the definition of probability, the notation: P(an event), simple problems such as pulling a blue marble out of a bag of marbles of several colors.
Briefly describe objectives of this lesson and the subject of the video.
FOCUS FOR VIEWING
To give students a specific responsibility while viewing the video, distribute the "Focus for Viewing" worksheets and specify expectations for completion. Alert students to the importance of drawing inferences, not merely "
answers
." They should expect frequent pauses, with time to complete each question on the worksheet.
VIEWING ACTIVITIES
Note: Students will be especially interested in the description of the launch failure and explosion at the beginning. You might be asked to rewind and show this segment a second time without interruption!
Start when you see Dr. Therese Amabile (blue sweater) saying, "It's fairly easy to see that things like flipped coins .... " Pause when you hear, "... a matter of life and death when the events in question are critical equipment failures."
Give students time to work on Question 1 on their worksheets and discuss question as necessary.
NOTE: The next segment shows the explosion and the simultaneous reactions of the NASA engineers. Prepare students for this serious subject matter.
Start video and pause when you see Dr. Bruce Hoadley, Bell Labs statistician (man with glasses) and hear, "... with no attempt to actually quantify the probabilities."
Give students time to work on Questions 2 and 3 and discuss questions as necessary.
Start video and pause when you hear Dr. Hoadley say, "... the probability of failure of a particular field joint was, we came out with, .023".
Give students time to work on Question 4 and discuss question as necessary.
Start video and pause when you see "Multiplication Rule ... " on the screen and hear Dr. Amabile say, "... is the product of their individual probabilities."
Give students time to work on Question 5 and discuss question as necessary.
Start video and pause when you see "First Flip P(H) = 1/2 Second Flip P(H) = 1/2 P(H,H) = 1/4" on the screen and hear (Dr. Amabile) say, "... one half times one half, or one fourth."
Give students time to work on Question 6 as discuss question as necessary.
Start video and pause when you see "(.977) (.977) (.977) (.977) (.977) (.977)" on the screen and hear Dr. Amabile say, "... so we have to multiply .977 times itself six times."
Give students time to work on Question 7 and discuss question as necessary.
Start video and pause when you hear Dr. Hoadley say, "... like playing Russian Roulette with eight bullets."
Give students time to work on Question 8 and discuss question as necessary.
Start video and pause when you hear Dr. Hoadley say, "... because of a redundancy in the O-rings."
Give students time to work on Question 9 and discuss question as necessary.
Start video and pause when you hear, "... the independence assumption was faulty."
Give students time to work on Question 10 and discuss question if necessary.
Start video and stop when you hear Dr. Amabile say, "... success or failure of any complex system."
POST-VIEWING ACTIVITIES
Simulation of Field Joint Failure
Discuss simulation: why and how to simulate the field joint problem. In groups of three (preferably), the following roles should be distributed: recorder, dice roller, number cruncher. Clarify role expectations for students.
Simulate one field joint's operation by tossing one die, and considering "success" as a roll of 1, 2, 3, 4, or 5. Thus, P(success) = 5/6 = .83 = 83%. Toss one die ten times, record success or failure, then tally all the rolls of the students on a chart at the front of the class. (The class average should approach 83%.)
Discuss the difference between individual groups' results and the class average. Does it approach 83%? (This demonstrates Probability's Law of Large Numbers.)
Discuss "failure" in the following simulation. (P(failure) = .17 = 17%.) Guide class discussion to awareness and understanding of P(failure) = 1 - P(success).
Toss six dice at a time, to simulate six identical parts working independently. "Success" is all six dice showing 1, 2 ,3, 4, or 5. Have groups throw six dice ten times, record success or failure, then tally the class data as before. Display on chart at the front of the class. (The class average should approach 33% "success.")
Discuss results. Make connection to Challenger data for field joints.
Important conclusions: empirical demonstration of (1) Law of Large Numbers, and (2) Multiplication Rule for Probability of Independent Events.
ACTION PLAN
To extend this lesson outside the walls of the classroom:
1) Invite engineer(s) to class to reinforce lesson with real-world examples.
2) Write to NASA (perhaps to Dr. Bruce Hoadley) for further information or data. Alternatively, write to local manufacturer of machine parts.
3) Design a more complex simulation, using hardware, electronics, etc. Engage assistance of community scientists or engineers. Share results with school or local community. Present results to other classes.
EXTENSIONS
Related subjects for interdisciplinary ideas: physics, electronics, metal or wood technology, computer science, automotive technology.
Web sites to investigate:
http://forum.swarthmore.edu/~steve/steve/probability.html
EXHAUSTIVE: all-level links to probability sites.
http://www.dartmouth.edu/~chance/
Materials designed to help teach an introductory probability/statistics course, using current news. Level: high school, college. Several links.
http://www.forum.swarthmore.edu/probstat/probstat.html
Links to classroom materials, software, projects, public forums.
http://www.psoup.math.wisc.edu/
Gorgeous downloadable graphics. Demonstrates the use of much higher probability at graduate school level. An award-winning site.
www.the-web.net/Lotto/
Maclean's Magazine article on probability and the Canadian Lotto.
http://ericir.syr.edu/Virtual/Lessons/Mathematics/Probability/index.html
Lesson plans, K-12.
www.mste.uiuc.edu:591/mathed/completelist.html
Math lessons database, K-12, from the University of Illinois at Urbana-Champaign.
FOCUS FOR VIEWING WORKSHEET
THE CHALLENGER EXPLOSION
(or; How NASA Played Russian Roulette)
Please follow the video and fill in the blanks. Most questions require you to draw inferences: watch thoughtfully !
(1) Independent events are events which
(2)Qualitative probability uses to describe the likelihood of an event.
(3)Quantitative probability uses to describe the likelihood of an event.
(4) If the probability of failure of a field joint is .023, P(success) =
(5) The Multiplication Rule states that if two events, A and B, are, the probability of both happening is the of their separate probabilities.
(6) P(H,H) when tossing two coins is x =
(7)P(success of six field joints) = (.977)6 , or approximately ______________________%.
(8)Dr. Hoadley makes a slight error when he says that 13% failure is "like playing Russian Roulette with eight bullets." What is he trying to communicate?
_______________________________________________________________________________ (9) The NASA engineers put "redundancy," or "back-up," into the O-ring of each field joint, by installing
. (10) The O-ring pairs were discovered NOT to be independent "events" because
.
:
"FOCUS FOR VIEWING" WORKSHEET
1) Do not influence each other. The outcome of one event does not affect the outcome of any other.
2) words or phrases
3) numbers or quantities
4) 1 - .023, or .977
5) independent, product
6) 1/2 x 1/2 = 1/4.
7) 87%
8) 13% is approximately 1/8. The chance of field joint failure is equivalent to the chance of killing yourself playing Russian Roulette with one bullet in an eight-chambered gun. The engineer is using a powerful image to convey his opinion of the higher risk of failure.
9) a second O-ring.
10) pressure from escaping gases caused the movement of one to affect the performance of the other; "joint rotation." They did not perform independently of each other.
