WHAT'S THE CHANCE?
In this lesson, students will develop a definition of probability. While solving problems of probability, students will have to calculate total number of possible outcomes of events. This will involve calculating the number of possible combinations of items. Students will develop a general rule for counting the number of combinations possible in various situations. The concepts developed in this lesson should be used to solve problems from a variety of contexts, such as insurance costs, genetics, and society. The lesson can take two class periods, with the second class period used to discuss the problems from part IV of the lab worksheet.
Math Talk #109: Let Me Count the Ways
Math Vantage: Data Analysis #15: Chances Are
Students will be able to:
1. calculate the probability of an event occurring.
2. calculate the number of combinations possible, given restrictions.
3. solve problems involving probabilities.
Texas Assessment of Academic Skills (TAAS), Exit Level
#5: Demonstrate an understanding of probability and statistics.
#11: Determine solution strategies and analyze or solve problems.
NCTM Standards for Grades 9-12:
Standard 3: Mathematics as Reasoning
Standard 11: Probability
Per group of 3-4 students:
- lab worksheets (1 per person)
- spreadsheet computer program with projection capability (optional)
- pair of dice
- 2 coins
- 10 sheets of yellow, red and green paper
- 10 strips of black, brown and blue paper
Have the room set up in groups of three to four. On each table there should be 2 coins and a pair of dice. Ask each group to discuss and answer the first question in part I of the Probability Lab worksheet. When the groups have had time to discuss the question, call on a group to report their answers and explain how they decided on each answer.
1. 50, since for each toss there is a 50% chance of a head coming up.
After discussing the first question, simulate a coin toss with a spreadsheet. Use of a computer is nice if the technology is available. Watch the bar graph for several "tosses" of the coin. Notice that even though we know there should be 50 heads out of 100 tosses, this does not always happen. Ask the students to explain this. (Accuracy is dependent on a large sample size.) Tell students that the questions we are talking about deal with the concept of probability. We say the probability of tossing a fair coin and getting heads is 1/2 or 50%. Ask a student to give a definition of probability based on our discussions. (probability = the ratio of the number of successes to the total number of possibilities) Using their definition, have each group discuss the remaining questions in part I. When the groups have had time to discuss the remaining questions, call on each group to report their answers and explain how they decided on each answer.
2. 10, since there are four possibilities - TT, TH, HT, HH. Two out of four of these have exactly one head, so 1/2 of the tosses will have at. 5, since one out of four have two heads. 14, since three out of the four possibilities have at least one head.
4. 5/36 (see chart below)
5. 7, with a probability of 1/6 (see chart below)
Sum chart for rolling a pair of dice
To give students a specific responsibility while viewing the video, tell students to watch for a variety of situations where calculating the probability is complicated by calculating the total number of possibilities. In the first video, we join "Super Guy," a local super hero, in the midst of a crisis.
BEGIN the Math Talk video when the screen is filled with the Water Works sign and the narrator is describing the problem of leaking water. PAUSE after Super Guy changes clothes and the salesman says, "You've got a problem." Ask a student what is Super Guy's problem. (save the city from flooding)
RESUME the video. PAUSE after the salesman says, "You forgot your cape." Say to the class, "Now Super Guy has another problem. How can he solve the problem of not having a cape?" (buy one at the store)
RESUME. PAUSE as Super Guy is putting the blue belt with the yellow cape and says, "...the blue belt with the yellow cape." Ask a student how we could calculate the probability of Super Guy buying the blue belt and the yellow cape. (find out the total number of possible combinations of capes and belts)
RESUME to see what Super Guy does. PAUSE after Super Guy says, "This is so confusing." Ask a student, "Why is Super Guy confused?" (He must decide on what color belt and cape and there are a lot of choices.).
RESUME. PAUSE after Super Guy says, "How many are there?" Ask the students to guess how many different combinations of belts and capes Super Guy has to choose from. [Accept all guesses. Do not validate or correct.]
RESUME the video to see how Super Guy and the salesman answer the question. PAUSE as the two citizens are trying to plug the water leaks and the narrator says, "Super Guy".
Say to the class, "It looks like Super Guy needs our help deciding how many choices he has. In your groups, use the colored paper and colored strips to decide the number of combinations possible for Super Guy's cape and belt." While the students are working, FAST FORWARD the video to the point where all nine combinations are on the display ordered by color and Super Guy has just pushed the salesman to the window. Cover the screen so that students do not see the solution before they finish in their groups. Ask the groups to share their answers and how they arrived at their solution. Uncover the screen to show how Super Guy and the salesman solved the problem. Ask the class if they notice anything about the number of capes, the number of belts and the number of combinations. <
B>RESUME the video to hear Super Guy's explanation. PAUSE after the salesman says "wonderful." Ask students to respond to Super Guy's question about multiplying the number of capes and belts to find the number of combinations. Then ask students to answer the original question: "What is the probability that Super Guy will buy the blue belt and the yellow cape." (1/9) So the chances are pretty slim that he will buy this combination.
RESUME to see what he chooses. STOP the video as Super Guy flies out the window and the narrator says, "Super Guy".
Say, "Let's investigate some other probability problems". EJECT the Math Talk video and INSERT the Math Vantage video already cued to begin after the various shots of cars when the host is standing by a convertible. BEGIN the video. PAUSE the video on the graphic of the possible choices of transmissions, colors and interiors. Ask students to write on a piece of paper the car they would buy. Ask a student to tell his/her choice to the class. Collect any other papers with the same choice and tape them on the board together. Continue this until everyone's choice is on the board. Ask student how many combinations of these items there are. Let the students discuss this in their groups, reminding them about Super Guy's strategy for solving to the cape problem. While students work, place a sheet of laminating film over the monitor screen. After they have discussed the problem in their groups,
RESUME the video and PAUSE immediately on the possible choices of transmissions, colors and interiors. Ask a member of one group to come to the TV monitor and, using a pen for overhead transparencies, draw all of the possibilities on the screen by connecting items from each category. Ask the class if all possibilities are represented.
RESUME the video to verify. PAUSE after "16 choices". Ask a student for the probability of the host buying an automatic red car with leather interior. (1/16) Clean the film on the TV monitor and
RESUME the video to verify the probability. PAUSE after "1/16" and ask students why this probability may not be true. (It may be influenced by personal preferences.)
RESUME. PAUSE after the host says, "Now all I need is insurance." Ask the class if anyone knows how insurance rates are set.
RESUME to find out. PAUSE after "Why?" as the police car pulls the red car to the side of the street. Ask for responses to the question posed: "Why do teenager and people who drive red cars pay more?"
RESUME to find out the answer. PAUSE after you hear, "...the more you should have to pay
Tell students, "Watch the next segment for another fashion nightmare."
RESUME the video. PAUSE after the host holding some clothes says, "3 pairs of shorts, 3 shirts and 2 hats." Ask the class to answer her question. (There are 18 possible outfits.)
RESUME. PAUSE after "...different from theoretical probability." Say, "Here the host talks about the same issue we brought up with the cars. Outside factors do have an effect on probabilities. Now let's look at telephone numbers."
Ask students to guess how many possible phone numbers there are in the US. Record student guesses on the board. Tell students to watch the next segment to see how close our guesses are.
RESUME. PAUSE after "...without making the telephone numbers longer." Ask students for possible solutions. Ask if order is a factor when putting numbers together to make a phone number. Have students listen to the next segment to learn what a permutation is.
RESUME the video. PAUSE after "permutation" while the word is still on the bottom of the screen. Ask someone to describe a permutation in their own words. (putting items together where order matters) Tell students to record this definition on their lab worksheet.
RESUME the video to find out how many possible phone numbers there are. PAUSE after "That's 800 choices." Ask a student why there are only 8 choices for the first number in the area code. (can't use 0 or 9) Ask another student to relate the method used for finding the number of possible area codes to the other problems we have solved today. Tell students that there are really fewer than 800 area codes. Ask if they can think of any reason for this.
RESUME. PAUSE after "..any one of ten different digits" while the arrow is moving across the last four digits. Ask a student how many possibilities there are for the final four digits in a phone number. (10x10x10x10=10000) Ask students how they can use these three answers (792, 800 and 10000) to find the total number of telephone numbers. (multiply them)
RESUME the video. PAUSE after "What are the chances this is my number?" as the host is standing by a huge telephone. Ask students to come up with an answer. To do this, we must multiply the three numbers. Check to see whose guess was closest to the total number of phone numbers. Then
RESUME the video to see the probability. PAUSE after the host reads the answer.
Summarize with students the method for finding the total number of combinations we have been using (multiplying the possibilities). Ask students if they can think of when this might not work. FAST FORWARD the video to the screen just after the "Ellen" license plate, as the truck drives to the front of the screen.
RESUME the video. PAUSE after "We're talking about a combination," while the word "combination" is still on the screen. Ask students to state the difference between a permutation and a combination. (In a combination, order doesn't matter.) Have students record this on the lab worksheet.
RESUME for a problem of combinations. PAUSE after "That leaves three choices for the place by the door." Ask students how many combinations they think there are.
RESUME to verify student answers. PAUSE after "12 possible arrangements of two people." Ask the class if they think this is correct. (no)
RESUME to find out why. STOP the video after "...taking only two at a time".
Tell the students, "The problem we just solved can be written in the following way: 4C2 which is read "4 choose 2". What if we change the problem to driving a car instead of a truck? Then we could carry 3 passengers, so the problem would be 4C3 (4 choose 3). How many combinations are possible?" Students should work in groups to solve this problem. Suggest that they use different colors of paper to represent a person so that they can model the problem like they did the Super Guy problem. They should come up with 4 different combinations. Tell them to show their answer on the lab worksheet. In their groups, the students should answer the remaining questions in part III of the lab worksheet. As the students finish the problems listed, discuss the general formula for finding the number of combinations. The formula is
nCr = n! __
Discuss the student's answers to questions 1-6 using the formula.
Answers - Part III.
Notice that aCa will always be 1 and bC1 will always be b.
Ask a student to summarize what they learned today. Ask other students to add to the student's summary. (It should include calculating probabilities, the difference between permutations and combinations, and calculating combinations.) Have students work in their groups to solve the problems in part IV. Assign each group the responsibility to explain one of the problems to the rest of the class if there are questions.
Answers to part IV.
4. Answers will vary. Possible solution: 3 shorts and 5 shirts
The class can create a survey to determine a driver's car color and number of accidents. The class should distribute the survey widely at school and in their neighborhoods. Students should investigate the insurance rates for the types of drivers surveyed. They can then analyze the results to see if the insurance companies are justified in charging higher prices for people who drive red cars.
Invite a guest speaker from the state lottery commission to discuss the probability of a person winning the lottery. The speaker can discuss the revenue earned from people playing the lottery and the uses of the funds. Before the speaker comes, students should determine the probability of winning the lottery.
Biology: Investigate the probabilities associated with genetics.
Chemistry: Investigate the number of ways to combine elements to create chemical compounds.
Geometry: Set up a geometric carnival, where students determine the probability of winning at games like shuffle board (uses the area of trapezoids and triangles to find probability), hitting the bulls eye in darts (uses the area of circles to find probabilities), or coin tosses (toss a coin on a grid without letting the coin touch a line of the grid).
1995-1996 National Teacher Training Institute / Austin
Master Teacher: Susan May
Click here to view the worksheet associated with this lesson.