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ProbabilitProbabilityy
Middle School Middle School Content ShiftsContent Shifts
Concerning Concerning probability, what probability, what have you usually have you usually taught or done? taught or done?
Share with an elbow Share with an elbow partner.partner.
Read “6 – 8 Statistics and Probability Read “6 – 8 Statistics and Probability Progression Document” pages 7 – 8. Progression Document” pages 7 – 8. Note anything that seems unfamiliar.Note anything that seems unfamiliar.
Common Common MisconceptionsMisconceptions
Students often expect the theoretical and experimental probabilities of the same data to match. By providing multiple opportunities for students to experience simulations of situations in order to find and compare the experimental probability to the theoretical probability, students discover that rarely are those probabilities the same.
Students often expect that simulations will result in all of the possibilities. All possibilities may occur in a simulation, but not
necessarily. Theoretical probability does use all possibilities. Note examples in simulations when some possibilities are not
shown. Melisa Hancock, KATM Flip Books
Cubes in a SackCubes in a Sack
With your partner, choose a paper sack. Do not look inside.Each paper sack holds 30 cubes. They may be in one of the following configurations:
Combination ACombination A25 25 red and 5 5 blue
Combination BCombination B20 20 red and 10 10 blue
Combination CCombination C10 10 red and 20 20 blue
25 Draws in All25 Draws in AllOne student will draw out a cube, note its color, and then return it to the sack. The partner will record which color it was.
After 10 drawsAfter 10 draws, compute the percentage of red cubes and then predict which sack you have: A, B, or CA, B, or C
After 20 drawsAfter 20 draws, compute the percentage of red cubes and then predict which sack you have: A, B, or CA, B, or C
After 25 drawsAfter 25 draws, compute the percentage of total red cubes and make a final prediction: A, B, or CA, B, or C
Write an Write an ArgumentArgument
With your partner, write an argument (based on your experiment and your
predictions) about which of these sacks you have:
Combination A: Combination A: 25 25 red and 5 5 blueCombination B: Combination B: 20 20 red and 10 10 blue Combination C: Combination C: 10 10 red and 20 20 blue
Go to Your CornersGo to Your CornersMeet with other
partnerships that shared your prediction.
Discuss your arguments.
Finally, check your sack to see if you were correct.
Conjecture…..
Using the combination in your sack, if you did 300 draws, how many times do you anticipate getting a red cube? Why?
SummarizeSummarizeDid any partnership predict a different combination than what they had?
What number of samples would have made you most confident in your prediction?
How did your group respond to the question about 300 draws?
Iowa Core StandardsIowa Core Standards7.SP.6.7.SP.6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. 7.SP.7. 7.SP.7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. a.Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
How Many Throws?How Many Throws?
Discuss with Discuss with your Partner….your Partner….
”If you were given 2 dice and were to roll them together, how
many different outcomes would be
possible?”
Before you begin…..Before you begin…..• One partner will roll the dice and state the One partner will roll the dice and state the
outcome.outcome.• The other partner will make a tally mark in the The other partner will make a tally mark in the
appropriate space.appropriate space.
Discuss with your partner how many throws it Discuss with your partner how many throws it will take to get at least one tally in every will take to get at least one tally in every space. Place that number at the top of the space. Place that number at the top of the sheet.sheet.
A red 3 red 3 and a green 2 green 2 total 5.
A green 3green 3 and a red 2red 2 total 5.
ButBut the tallies go in different spaces on the recording sheet.
Roll!
Continue rolling and recording until all spaces have at least
one tally.
When you are finished, count up the tallies and compare
them to your estimate.
Tell your teacher when you are done.
Looking at the ResultsLooking at the Results
With your partner, record all the statements you can make about the sums of the two dice. Be prepared to share one with the entire group.
SummarizeSummarizeCould the chart have been filled in fewer rolls? Explain.
What if we combined our class data? How would you feel about your statements about the sums of the dice?
How many rolls would have made you feel confident in your statements?
Iowa Core StandardsIowa Core Standards
7.SP.6. 7.SP.6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
How to PlayHow to PlayEach trio will decide who will be Player APlayer A, , Player BPlayer B, and , and Player CPlayer C..
All players make a fist and on the count of four, each player shows one of the following:•rock (by showing a fist)•scissors (by showing two fingers)•paper (by showing four fingers)
Play 20 RoundsPlay 20 Rounds
Player A gets a point if all players show Player A gets a point if all players show the same sign.the same sign.
Player B gets a point if only two players Player B gets a point if only two players show the same sign.show the same sign.
Player C gets a point if all players show Player C gets a point if all players show different signs.different signs.
Is this game fair?Is this game fair?
If you were to play the If you were to play the game again, which game again, which player would you player would you rather be? Why?rather be? Why?
How can you make How can you make this a “fair” game?this a “fair” game?
SummarizeSummarizeUnder the original rules, is one of the players more likely to win than the others? How do you know?
How did you alter the rules to make this game “fair”?
If you decided to use Sheldon’s method, how many players would you need and what might be the rules?
Iowa Core StandardsIowa Core Standards7.SP.8.7.SP.8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. a.Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. b.Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event. c.Design and use a simulation to generate frequencies for compound events.
““Let it Snow”Let it Snow”Adapted from “Newspaper Route”, Navigating through Probability in Grades 6 – 8,
NCTM
LaunchLaunchWhen Chuck was younger he
had a paper route. Each customer paid Chuck $6 every week.
One day, Mr. Jones did not have the correct amount of cash for Chuck. He only had three $1 bills and one $20 bill.
He offered to let Chuck have one of the bills pulled out of a paper sack. Chuck was skeptical and said “no”. Finally, Mr. Jones wrote Chuck a check. However, on his way home, Chuck began to wonder, if he were to do this every week, would his chances of getting that $20 bill be worth the risk.
What advice would you give Chuck? Why?What advice would you give Chuck? Why?
ExploreExploreImagine that you are responsible for the snow removal for your elderly and wily neighbor, Miss Giving. She pays you weekly during the snow season (15 weeks from December to March) so as to budget her payments.
She would like to make you a deal. Instead of your weekly
$25 payment, Miss Giving would like to offer you a
chance to draw two bills, one after the other, from a bag
containing one $50 bill and five $5 bills.
Should you take her up on this offer? Or do you have
“misgivings”?
Complete the “Let it Snow” activity pages Complete the “Let it Snow” activity pages in your group.in your group.
SummariSummarizeze
What simulation did you use?
Is Miss Giving’s offer a good deal? Explain using the data you collected during your simulation.
What is the theoretical expected value of this scenario and how did you find it?
Iowa Core StandardsIowa Core Standards7.SP.6.7.SP.6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. 7.SP.7. 7.SP.7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. a.Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.b.Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
Iowa Core StandardsIowa Core Standards7.SP.8.7.SP.8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. a.Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. b.Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event. c.Design and use a simulation to generate frequencies for compound events.
One and One
Equals Win
The SituationThe SituationThe basketball team is down by one with one
second on the clock but a foul may save them.
Can a 60% free throw shooter save the game for
her team? How often?
Simulate the SituationSimulate the Situation
Using technology, perform a simulation that will determine how often a 60% shooter can win the
game.
Use the Common Core Tools or a graphing calculator.
The The SummarySummary
What connection is there between the free throw shooter’s percentage and the percent wins?
What percentage of times did the shooter lose the game? What does that have to do with the shooter’s percentage?
Iowa Core StandardsIowa Core Standards7.SP.8.7.SP.8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. a.Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. b.Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event. c.Design and use a simulation to generate frequencies for compound events.
Probability TalkProbability TalkTwo dice were rolled and their sum was graphed. The rolling continued until one sum occurred ten times.
During the rolling, a horizontal line was placed on the first sum that occurred:Two timesFive timesEight timesTen times
What does your data tell What does your data tell you…you…
Number of throws needed to get one sum tenten times
Sums that first occurred
‒ two timestwo times
‒five timesfive times
‒eight timeseight times
‒ten timesten times
Any Any conclusionsconclusions
Or patterns?Or patterns?
