Calculating Chances: What is the Probability of an event? By Christa ...

1Calculating ChancesWhat is the Probability of an event?By: Christa PipoGrade: 7 - 95 day Unit Plan

6Name _____________________________Data Collections SheetRoll Dice 1 Dice 2 Total1234567891011121314151617181920Class DataDice Totals:Flip1234567891011121314151617181920Coin1Coin Flips:2 _______________ HHH _____________3 _______________Coin24 _______________ HHT ______________5 _______________6 _______________ HTT _______________7 _______________8 _______________ TTT _______________9 _______________10 _______________11 _______________12 _______________Coin3Outcomein HTformat

7Name __EXAMPLE____________________Data Collections SheetRoll Dice 1 Dice 2 Total1 3 5 82 1 1 23 5 2 74 1 5 65 4 6 106 5 4 97 6 6 128 4 3 79 4 5 910 1 5 611 1 2 312 6 4 1013 4 6 1014 2 3 515 3 2 516 6 4 1017 5 4 918 2 6 819 1 1 220 3 3 6Class DataDice Totals: (Theoretical)FlipCoin1Coin2Coin3Outcomein HTformat1 H T H HTH2 H H T HHT3 T H T THT4 T T T TTT5 T H T THT6 H H H HHH7 T T H TTH8 H T T HTT9 T H T THT10 H H H HHH11 T H H THH12 T H H THH13 H H T HHT14 T T H TTH15 H H T HHT16 H T T HTT17 T H H THH18 H T T HTT19 H H H HHH20 H T H HTHCoin Flips: (Theoretical)2 ____5_________ HHH _____15______3 _____9________4 ______15_______ HHT _____50_______5 _____22________6 _____30_______ HTT _____48________7 _____40________8 _____32________ TTT ______17_______9 _____25________ They should see that the numbers at the extremes are less likely10 ____12_________ than the middle numbers. With the coin flips they should see that11 ____7__________ HHH and TTT are less likely than HHT or TTH12 ____4__________77

Lesson 2Learning Outcomes: The students will be able to find the chances of a particular outcomefor some events as a fraction and they will be able to give a general description of whysome outcomes are more likely then othersAnticipatory Set: Have you ever thought about your chances of getting 2 ones when youroll a pair of dice? What about the chances of rolling a 7 or 8? Do you have better chanceof one than the other? Why?Sharing the Goals:Today we are going to learn how to figure out how to find the chances ofgetting a certain outcome with 2 dice from a table or using a tree diagram.Instructional Presentation:1. Break the students into the same groups as the previous class and have them getout pencil, paper, and the data they collected.2. For the dice rolls have them figure out how many outcomes add to each total,2-12, filling in the table as they go.3. Have them use the tree diagrams to find the chances of each possible outcomeas a fraction and then compare these fractions to the classroom data.Guided Practice:1. Have the students finish filling in the table of possible outcomes and find thechances of getting particular outcomes on a roll of 2 dice.2. Show them how this relates to the tree diagram and how they can use either thetree diagram or the table to get the probability of a certain outcome in terms of afraction.8Homework: Have them fill in the table for the coin flips and use it to find theprobability of each outcome as a fraction.Have them think of one other area where they use or see probability.

Name _______________________________9Outcome(Roll of2 dice)Possibility1Possibility2Possibility3Possibility4Possibility5Possibility6TotalPossibilities2 1,1 13 1,2 2,1 2456789101112TotalPossibleOutcomesOutcome (3coin flips)3 Heads2 Heads/1 Tail1 Head/2 Tails3 TailsPossibility 1 Possibility 2 Possibility 3 TotalPossibilitiesChances of thisoutcomeTotalPossibleOutcomesTotal =

Name __Answer Key________10Outcome(Roll of2 dice)Possibility1Possibility2Possibility3Possibility4Possibility5Possibility62 1,1 13 1,2 2,1 24 1,3 3,1 2,2 35 1,4 2,3 3,2 4,1 46 1,5 2,4 3,3 4,2 5,1 57 1,6 2,5 3,4 4,3 5,2 6,1 68 2,6 3,5 4,4 5,3 6,2 59 3,6 4,5 5,4 6,3 410 4,6 5,5 6,4 311 5,6 6,5 212 6,6 1TotalPossibleOutcomesTotalPossibilities36Outcome (3 Possibility 1 Possibility 2 Possibility 3 Total Chances of thiscoin flips)Possibilities outcome3 Heads HHH 1 1/82 Heads/ HHT HTH THH 3 3/81 Tail1 Head/ HTT THT TTH 3 3/82 Tails3 Tails TTT 1 1/8TotalPossibleOutcomes8Total = 8/8Some answers could be when playing game or gambling. The chance of getting caughtdoing something or that they might meet somebody are just some examples. There arealmost infinite possibilities that count as long as they actually relate to probability.Outcomes as fractions2 1/36 6 5/36 10 3/36 = 1/123 2/36 = 1/18 7 6/36 = 1/6 11 2/36 = 1/184 3/36 = 1/12 8 5/36 12 1/365 4/36 = 1/9 9 4/36 = 1/9

Lesson 3Learning Outcomes: The students will be able to find probabilities in terms of fractions,decimals, and percents and they will be able to compare the probabilities of unrelatedevents.Anticipatory Set: Which outcome is more likely, getting a 9 on a roll of 2 dice or drawingan ace or king from a standard deck of 52 cards? How can you figure this out?Motivational transitioning: What numbers are easier to compare, fractions, decimals, orpercents? How can you tell if one outcome is more likely than another?Sharing the Goals: In this class we will be thinking about possible outcomes whendrawing cards from a standard deck and we will be finding ways to compare theprobabilities of different events.Instructional Presentation:1. Have the students think about the chances of drawing a particular card or acertain type of card from a standard 52 card deck (Face card, a card of a certainsuit, etc.).2. Have them think about how they can compare the chances of drawing a certaincard to getting a certain number on a roll of 2 dice.3. Show them it is easier to compare decimals or percents than fractions.Guided Practice:1. Break the students into their regular groups and find the probability of drawinga certain type of card.- Give each group different criteria for the draw.2. Have the students use the previous data to find the chances of getting a numberon a roll of 2 dice.- Once again give each group a different number.3. Have them compare their outcomes and guess which one is more likely.4. Show them that they can covert the outcomes into decimals and percents andthen compare them more easily.5. Have them add the fractions, decimals, and percents of each possible outcome.Show them that the fractions and decimals will add to 1 and the percents to 100%.(The coin flips are useful as an example because there are fewer possibleoutcomes)11

Homework:Have them convert the probabilities from the dice roll into decimals and percentsand then have them compare the chances of a particular outcome with a certaincard draw (chosen at random) that also must be calculated in terms of a fraction,decimal and percent.Which dice rolls are more likely? Less likely?12Give them an example of a bag with a certain set of cubes and have them find theprobability of drawing a certain colored cube.Have them think about the probabilities of what will be drawn the second andthird times both if the cube is replaced and if it is not replaced by drawing treediagrams and tables for both situations.Tell them to think about what the probability would be of rolling a 6 with a singledice and getting a head on a single coin flip at the same time. Draw two treediagrams, one starting with the coin, the other starting with dice. Are the possibleoutcomes the same? Why?

Homework Answer Key13Outcomes as fractions, decimals, and percents.DecimalPercent2 1/36 .0278 2.78%3 2/36 = 1/18 .0556 5.56%4 3/36 = 1/12 .0833 8.33%5 4/36 = 1/9 .111 11.1%6 5/36 .139 13.9%7 6/36 = 1/6 .167 16.7%8 5/36 .139 13.9%9 4/36 = 1/9 .111 11.1%10 3/36 = 1/12 .0833 8.33%11 2/36 = 1/18 .0556 5.56%12 1/36 .0278 2.78%An example of the card draw would be a black king or a jack.Card draw = 6/52 = 3/26 = .115 = 11.5%There is a better chance of rolling a 6, 7, or 8 o a pair of dice than getting one of thesecards out of a deck because the decimal (or percent) probabilities of these outcomes isgreater than 11.5%.There is a better chance of drawing one of the specified cards than getting a 2, 3, 4, 5, 9,10, 11, 12 because their probabilities are lower than 11.5% or .115.

Answer key continued.A bag with 2 red cubes and 2 green cubesTree DiagramDraw 1 – With ReplacementRG14R G R GR G R G R G R GDraw 2 – Without ReplacementRGR G R GG R G R G RThere are 8 possible outcomes in the first example, but only 6 in the second examplebecause if you don’t replace the cubes by the third draw 2 outcomes from the firstexample cannot happen because either both green or both red cubes have already beendrawn.Therefore the possible outcomes in the first example are: RRR, RRG, RGR, RGG, GRR,GRG, GGR, or GGG. In the second example the outcomes could only be RRG, RGR,RGG, GRR, GRG, or GGR because there are only two of each color so you cannot getRRR or GGG.HT1 2 3 4 5 6 1 2 3 4 5 61 2 3 4 5 6H T H T H T H T H T H TThey should see there are 12 possible outcomes both ways and they are the sameoutcomes except for the order they happen.

Lesson 4Learning Outcomes:The students will be able to find the total number of outcomes possible with 2 or 3unrelated events with different numbers of individual outcomes such as a numberon a dice.They will be able to find the probability of a particular outcome of a combinationunrelated of events.Anticipatory Set: We have been working on events with equal possibilities, but what ifthe events have different numbers of possibilities? Can we still find the chances of anyparticular outcome?Motivational transitioning: What are the chances that if you toss a coin and through adice you will get a tail and a 4. How many possible outcomes are there with a dieand a coin? What about 2 dice and 1 or 2 coins?Sharing the Goals:Today we are going to find out how to answer these questions and useprobabilities we have found before to figure out the chances of certain outcome.Instructional Presentation:1. Have the students break into their regular groups.2. Use a tree diagram to show the students how to find the total number ofpossible outcomes for one die and a coin toss to keep it a limited number.3. Show them that as in previous cases they need to multiply the outcomes of thefirst event with the outcomes of the second event to get the total number ofpossibilities.4. Have the groups find the total number of outcomes with 2 dice and 2 coin flips.Then have the students find the probability of getting a 7 and 2 heads (or anotheroutcome)Guided Practice:1. Have the groups find the total number of outcomes with 2 dice and 2 coin flips.Then have the groups find the probability of getting a 7 on the dice and 2 headson the coins (or another outcome).2. Give them a model of a bag and colored cubes with different numbers of eachcolor and then have them find the probability of drawing a particular color.3. Have the groups find the probability of drawing a certain colored cube andgetting a certain number when they roll 2 dice. Give the different groups differentoutcomes.4. Have each group put their answers on the board with explanations.15

5. Talk about how probability is used on different games.-Can they think of other ways they have used probability in a game –spinners or different colored squares having different meanings are possibilities.6. Show them how the areas of a spinner relate to fractions and have them find thechances of landing on a certain color given the number of colors and telling themthe areas are the same size.7. Ask them to find the answer if there are 3 colors where the colored areas arenot equal on size. An example would be _, _, and _ of the spinner instead of eachsection being 1/3 of the spinner.8. Give the groups a few minutes to come up with the general outline of a gameinvolving probability.Homework:Have the students work on the games their groups came up with. The majorcriteria is that the need to be able to calculate the probability of getting a good orbad outcome on each turn.16

17Lesson 5This could take one or two days depending on the complexity of the games.Materials: Poster board and markers for each group.Learning Outcomes: The students will demonstrate they can find the probability of agiven outcome and be able to use that to help decide their moves in a game.Sharing the Goals:Today the groups will use their ideas from homework to come up with a gamethat allows them to calculate the probability of a good or bad outcome for eachmove.Instructional Presentation:1. Have the students break into their groups and start working on the game andgive them 5 minutes to combine their ideas.2. Give each group a poster board, markers, and dice, cards, coins, or othermanipulative they need to play their game.3. After the games are set up have the students play in their groups with thecondition they need to calculate the probability of a good and/or bad outcome.4. Each group presents their game to the class.Homework:The students need write up how they found their chances of winning or getting agood outcome on a move and how they can improve the game.Grading the game.The students get a group grade based on these criteria:1. Creativity2. How random are the chances of winning – everybody should have an equal chancewhen the game starts.3. How many different ways did they use probability.