What s the Probability I Can Draw That? Janet Tomlinson & Kelly Edenfield

Engage Your Brain On your seat you should have found a list of 5 events and a number line on which to rate the probability of those events occurring in your life. Turn to your neighbors and compare your answers.

Engage Your Brain A. I would get 2 chairs to turn for me on The Voice. B. It will snow in my hometown town before the end of the month. C. I will go on a well-deserved summer vacation. D. I will be teaching the same grade/classes next year. E. I will spend at least 3 days enduring enjoying professional development this summer.

Session Objectives Represent simple and compound probability problems by creating charts and diagrams. Connect intuitive ideas about probability with fractions and percents. Use diagrams to develop solution strategies for solving simple and compound probability problems.

Standards Related of Probability Grade 7 Investigate change processes and develop, use, and evaluate probability models. Probability is between 0 and 1. Approximate probability by observing long-run frequencies. Develop probability models for uniform and non-uniform probabilities. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Grade 8 Use relative frequencies to describe associations in bivariate categorical data.

Standards Related of Probability High School Understand events as subsets of a sample space Understand independence and conditional probability Use two-way tables to decide if events are independent and to approximate conditional probability Recognize and explain independence and conditional probability in everyday situations. Use the rules of probability to compute probabilities of compound events.

Probability Defining Probability Intuition about Probabiilty

Defining Probability What do you think of when you hear the word probability? How would YOU define it?

Defining Probability The ratio of the number of outcomes favoring an event to the total number of possible outcomes. The numerical chance that a specific outcome will occur. A branch of mathematics that measures the likelihood that an event will occur. Probabilities are expressed as numbers between 0 and 1. The probability of an impossible event is 0, while an event that is certain to occur has a probability of 1.

Using Probability to Make Decisions We do it all the time What do you think about when you re driving? Which way do I stand the best chance of NOT running into traffic? What s the chance I can make it under that yellow light before it turns red? I wonder if my friendly patrolman is sitting there waiting for me? Should I have packed a raincoat?

Students typically have an intuitive sense of probability. What do we do to draw upon that?

Using Intuition and Number Lines Connecting to Fractions and Percents Using Student Intuition about the World

A. A volcano will erupt right here in the next 5 minutes. B. I will go straight home when I leave school. C. Chase will always love to hunt. D. Ms. Bauer will be a bad teacher. E. Megan will be a professional flute player. F. Colby will do a flip-kick on his skateboard.

A number line can help student assign values to their intuitive probabilities. Begin with what they know and then add the mathematics. 0 ½ 1 0 0.5 1 0 50% 100%

Ask a Student to Decide You re planning the school field day. It s typically held outside but now there s a chance of rain. Decide: Are you going to move everything inside if there is a 90% chance of rain? 10% chance of rain? 75% chance of rain? 25% chance of rain? 40% chance of rain? 60% chance of rain?

Determining Probabilities We often jump into formulas for determining probabilities before fully developing understanding of the problem. Students also need ways to visualize probabilities. Visualizing the sample space is important for understanding the numbers that are used as our numerator as well as our denominator. Don t simplify too quickly.

Visualizing Sample Spaces Dice, Coins, and Spinners Tree Diagrams and Tables

Where do we usually start? Rolling dice Flipping coins Spinning spinners Drawing items from a bag

Creating a Sample Space Rolling a die

Creating a Sample Space Flipping a Coin Spinning a Spinner

If you are rolling a 6- sided die, what is P(6)? P(odd prime)? P(perfect square)? P(6) = 1/6

If you are rolling a 6- sided die, what is P(6)? P(odd prime)? P(perfect square)? P(6) = 1/6 P(odd prime) = 2/6

If you are rolling a 6- sided die, what is P(6)? P(odd prime)? P(perfect square)? P(6) = 1/6 P(odd prime) = 2/6 P(perfect square) = 2/6

Compound Events Determine the following probabilities. 1. Flipping a coin twice (or flipping two coins), determine P(at least one T). 2. Flipping a coin and rolling a 6-sided die, determine P(heads or 6). 3. Spinning a spinner with 4 equal sections, (red, blue, green and yellow) and rolling a 5-sided die, determine P(red or blue or even).

How can students solve these problems without using a formula?

Flipping 2 Coins

Flipping 2 Coins Determine each probability: P(Heads, Heads) P(at least one Tail) P(exactly one Tail) P(Head or Tail)

Flipping 2 Coins Determine each probability: P(Heads, Heads) = 1/4 P(at least one Tail) P(exactly one Tail) P(Head or Tail)

Flipping 2 Coins Determine each probability: P(Heads, Heads) P(at least one Tail) = 3/4 P(exactly one Tail) P(Head or Tail)

Flipping 2 Coins Determine each probability: P(Heads, Heads) P(at least one Tail) P(exactly one Tail) = 2/4 P(Head or Tail)

Flipping 2 Coins Determine each probability: P(Heads, Heads) P(at least one Tail) P(exactly one Tail) P(Head or Tail) = 4/4

Flipping a Coin AND Rolling a Standard Die Determine each probability: P(Heads) P(6) P(Heads and 6) P(Heads or 6) You roll a 6. Now, determine P(Tails and 6)

Flipping a Coin AND Rolling a Standard Die Determine each probability: P(Heads) = 6/12 (= 1/2 ) P(6) P(Heads and 6) P(Heads or 6) You roll a 6. Now, determine P(Tails and 6)

Flipping a Coin AND Rolling a Standard Die Determine each probability: P(Heads) P(6) = 2/12 (= 1/6 ) P(Heads and 6) P(Heads or 6) You roll a 6. Now, determine P(Tails and 6)

Flipping a Coin AND Rolling a Standard Die Determine each probability: P(Heads) P(6) P(Heads and 6) = 1/12 P(Heads or 6) You roll a 6. Now, determine P(Tails and 6)

Flipping a Coin AND Rolling a Standard Die Determine each probability: P(Heads) P(6) P(Heads and 6) P(Heads or 6) = 7/12 You roll a 6. Now, determine P(Tails and 6) Students often get 8/12 when they fail to look at the sample space; they simply add 6/12 + 2/12 and forget to subtract P(H and 6).

Flipping a Coin AND Rolling a Standard Die Determine each probability: P(Heads) P(6) P(Heads and 6) P(Heads or 6) You roll a 6. Now, determine P(Tails and 6) = 1/2

Your Turn! Draw a sample space for flipping a coin and drawing from a bag that holds 2 green, 2 blue, and 3 yellow marbles.

Flipping a Coin and Drawing Out a Marble Determine each probability P(Tails and Blue) P(Tails or Blue) P(Heads or Blue or Green) P(Heads and Yellow Yellow)

How Would You Draw the Sample Space? Flip a coin TWICE and spin a spinner with four equal sections of blue, yellow, red, and green.

Use the Diagram to Determine Each Probability P(Head, Head, Red) P(at least one Head and Yellow) P(Yellow or Blue) P(Head and Green Green)

Sample Space from Numerical Information The chance of rain in your hometown today is 10%. The chance of rain tomorrow is 70%. What is the probability it will rain on both days? What is the probability it will rain just one day? How can we represent this sample space?

The chance of rain in your hometown today is 10%. The chance of rain tomorrow is 70%. P(rain both days) P(rain only one day)

Building Models Have students work backwards from the probability to the sample space.

Working Backwards, #1 I m drawing marbles from a bag with the given probabilities. Determine the least number of marbles of each color that could be in my bag. P(blue) = 1/8 P(green) = 1/2 P(red) = 1/4 P(orange) = 1/8

Working Backwards, #1 P(blue) = 1/8 P(green) = 1/2 P(red) = 1/4 P(orange) = 1/8 O

Working Backwards, #2 Determine the probability of each color using the given information. P(blue) = 2 * P(red) P(green)= 1/2 * P(red) P(yellow) = P(blue)=3 * P(orange) Hint: build the sample space

Working Backwards, #2 Determine the probability of each color using the given information. P(blue) = 2 * P(red) P(green)= 1/2 * P(red) P(yellow) = P(blue)=3 * P(orange)

Working Backwards, #2 P(blue) = 2 * P(red) P(green)= 1/2 * P(red) P(yellow) = P(blue)=3 * P(orange) P(blue) = 12/37 P(red) = 6/37 P(green) = 3/37 P(yellow) = 12/37 P(orange) = 4/37

Why Use Models? Models can help students develop the basic rules of probability. Models give students a way to check answers if they are unsure of a rule. Models help students develop understanding basic probability without relying on formulas. Models give students a way to approach a problem if the don t remember a set of rules.

Frequencies

Using Survey Data Out of 75 freshman, 40% reported they did not like school lunches. 60% of the 85 sophomores reported they did not like school lunches. Only 20% of the 80 juniors reported they did like the school lunch. Represent this sample space in a chart.

One Possible Representation Why do you think I included the totals for the columns and rows?

A student is chosen randomly from this group. Determine the following probabilities. P(freshman who like lunch) P(the student does not like lunch) P(the student is a senior the student does like lunch) P(the student likes lunch or the student is a junior) P(the student is a senior)

Moving to Formulas How do you use these types of problems to develop and understand formulas? Let s look back at some of our problems and see where the formula fits in

Janet Tomlinson, jtomlinson@carnegielearning.com Kelly Edenfield, kedenfield@carnegielearning.com http://www.carnegielearning.com/nctm2017