©2013 Judo Math Inc.

CHANCE

©2013 Judo Math Inc.

7th grade

Statistics Discipline: Blue Belt Training

Welcome to the Blue Belt – What are the odds that you will do very well on this belt? 100%! And what are the odds that you will love probability? 100%! And what are the odds that a bird will fly over your head and leave a splat on your head at lunch today? Probably something like 0.1% (I hope that really doesn’t happen to you!) This belt is all about odds and probability… something that probably comes up in your life and conversation often. How many times have we all said, “what are the odds of ________”. In this belt we will be learning many different ways of answering such questions and how to mathematically represent probability.

Good luck grasshopper.

Standards Included:

7.SP.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

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 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.

o (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.

Order of Mastery: Calculating and approximating probability

1. Review fractions on the number line & conversions 2. Probability between 0 and 1 (7SP5) 3. Model vs. observed frequencies (7SP6 & 7) 4. Uniform and non-uniform models (7SP7)

1

1. Review: Fractions on the number line and convert

In order to become a probability expert, it is important to be up on your fractions, decimals, and percents work (see, we told you those things would come in handy!) In this short section, we are going to take a few minutes to review some work with fractions, decimals, and percents.

1. On the number line below, fill in as many possible numbers that you can between 0 and 1. Please try to put them in the correct place as well!!! (try to get at least 16)

a. How many numbers did you find between 0 and 1?

b. Do these numbers have anything in common with one another? What?

2. Now we are going to practice converting between fractions, decimals, and percents. Remember, if you forgot how to do any of these, feel free to look it up so that you complete this table as accurately as possible! Complete the table:

Fraction Decimal Percent

1/8

1/3

3/7

0.12

0.9

85%

57%

2

2. Probability between 0 and 1 (7SP5) The sock dilemma: There are 12 pairs of socks in Jamie’s drawer. 9 are red, 2 are blue, and 1 is green. She is getting dressed in the dark so she can’t see which color of socks she is reaching in and grabbing out. She is hoping to wear red socks today. Describe in words and with numbers (if possible) the likelihood of Jamie picking out that pair of red socks? If you can, describe the likelihood of picking the other colors too!

3

Probability is the likelihood that an event will take place. It can often be calculated by creating a ratio that is between 0 and 1 (like all the fractions on your number line at the front of this packet!) There is some important vocabulary that will come in handy:

Experiment A situation involving chance or probability that leads to results called outcomes

Outcome The result of a single trial of an experiment

Event One or more outcomes of an experiment

Probability The measure of how likely an event is

In order to measure probabilities, mathematicians have come up with the following “formula” where “A” is the desired outcome: So in the example with the socks, this would look like:

P(red)=9 𝑟𝑒𝑑

12 𝑡𝑜𝑡𝑎𝑙 =

3

4 OR the probability of picking red socks is ¾ or 75%

You can also represent the ratio on a number line, just like we practiced in part 1.

But what does this mean?!?! Probabilities can be understood on a continuum, or number line, from 0 to 1. Check out the line below

and then say what a ¾ probability indicates above…

Impossible Events Certain Events

Pizzas raining from the sky You come up with two…

_____________________________ _____________________________

The sun will rise in the morning You come up with two…

____________________________ ____________________________

3

4

4

1. Using the language from above, describe in words, the likelihood of each of these probabilities: P(event)= 1/8 P(event)= 1/12 P(event)= 7/12 P(event)= 3/4 2. What is the probability of choosing a vowel from the alphabet? Calculate the probability (should be a number between zero and one!) and then describe this probability using words. 3. A number from 1 to 11 is chosen at random. What is the probability of the number being odd? Calculate the probability (should be a number between zero and one!) and then describe this probability using words.

5

Quick Quiz 4. A bag has 40 blue chips and 60 green chips. What is the probability that you will randomly pick a blue chip from the bag? 5. A bag has 4 blue marbles, 5 red marbles and 6 green marbles. What is the probability of picking a red or blue marble 6. A single 6-sided dice is rolled. What is the probability of each outcome? What is the probability of rolling an even number? Of rolling an odd number? P(1)= P(2)= P(3)= P(4)= P(5)= P(6)= These are a little trickier… P(even)= P(odd)=

6

3. A spinner has 4 equal sectors colored yellow, blue, green, and red. After spinning the spinner, what is the probability of landing on each color?

P(yellow)= P(blue)= P(green)= P(red)= How would these probabilities change if the size of the red and yellow sections were increased by ½ and the size of the blue and green sections increased by ½? Draw the new spinner and calculate all probabilities.

7

3. Theoretical and experimental probabilities (7SP7) Calculating probability is essentially a mathematical estimate of the odds of something happening. That’s why the calculated probability is also called the “theoretical probability”. But if you were to model these situations (to determine the experimental probability), the model might not match the mathematical estimate. For example, the odds of flipping a coin and having it land on heads is ½. If you flip it again, the odds are ½ again. Even if you flip it 10 times, the odds will be 5/10 or ½! In reality, however, it could be the case that you flip a coin 10 times and get 10 heads! The theoretical probability may not match the experimental probability. In this section we will engage in some activities to test this out… Activity 1: Dice Rolling Step 1: Roll two dice 10 times.

Step 2: After each roll, note whether any sixes were observed and record your results in the table below.

Roll Any Sixes?

(Y/N)

1

2

3

4

5

6

7

8

9

10

a. What proportion of the 10 rolls resulted in at least one six?

8

b. Combine your results with those of your classmates. What proportion of all the rolls in the class resulted in at

least one six?

c. Make a list of all the different possible outcomes that might be observed when two dice are rolled. (Hint:

There are 36 different possible outcomes.)

d. What proportion of the 36 possible outcomes result in at least one six?

e. Suppose you and your classmates were able to roll the two dice many thousands of times. What proportion of

the time would you expect to roll at least one six?

9

2. Activity #2: Coin Flipping a. Flip coin 15 times. b. Record your results using a table similar to the one in the last activity. c. Using your results, create two different ratios representing the number of heads and tails over total attempts (15). d. With your knowledge of probability, write a 3-5 sentence paragraph describing your results and if they are what you expected. Try to use words like experimental and theoretical probability in your writing. If there was a discrepancy between these two probabilities reflect on why this might be the case!

10

Activity 3: Favorite Colors Gather some data on favorite colors of people in your classroom. Ask 12 people their favorite color and record it on the chart below.

Red

Orange

Yellow

Green

Blue

Purple

Pink

Black

Other

a) Based on the survey, what is the probability that the next person will say “blue”?

b) Based on the survey, what is the probability that the next person will say “purple”?

c) Based on the survey, what is the probability that the next person will say

“yellow”?

d) Based on the survey, what is the probability that the next person will say “orange”?

Continued…

11

e) Now ask 12 more randomly selected people. How do your probabilities match up from your two different samplings?

Red

Orange

Yellow

Green

Blue

Purple

Pink

Black

Other

New probabilities (compare these to old and explain in writing how they compare! Why might they be different or the same?)

12

Now try some practice problems with experimental and theoretical probability: 1. In a survey, Scott asked 50 people if they voted for the current president. He recorded the results in this table:

Yes No Not sharing this info

22 20 3

Based on the survey, what is the probability that the next person will say “No”? Justify your answer with at least 2 sentence and some numbers.

2. Your (not very bright) friend declares that since the likelihood of flipping a coin is ½ for heads, than the likelihood of a thumbtack landing on its point is also ½. Explain to your friend why this is not true.

13

3. Your teacher decides that he wants to get his students to like him more by doing a raffle each day in class for a free homework pass! If a student is selected at random from a class, find the probability that you will be selected. Suppose that you are not selected for 5 days in a row and all of the people that are selected are taken out of the running. What is yoru probability of winning

4. THE GREAT TESTING DEBATE: Your teacher decides to give you a multiple choice test where each question has the choices a, b, c, and d. You are not feeling prepared for the test so you RANDOMLY pick answers to each of the questions. What do you think the likelihood of getting each question right it? What grade do you think you will get on the test if you do this?

Create some arguments for and against answering questions randomly on the test. You may get the opportunity to engage in a debate in this for class!

Arguments FOR random answers Arguments AGAINST random answers

14

4. Uniform and non-uniform models (7SP7) A few final vocabulary words to add to your probability repertoire!

Probability Model The probability model is the set of all possible outcomes in a situation.

Uniform Model A model where all outcomes have the same likelihood of occurring (like a spinner with equal sections.)

Non-uniform Models: A model where there is not equal probability for all outcomes (like a spinner with UN-equal sections.)

1. In this UNIFORM MODEL, compare the theoretical and experimental probabilities for

the spinner landing on Green. Are they the same or different? Why or why not?

2. After 6 trials of rolling a dice, Chris found that the experimental probability for rolling a 4 was ½. IS a dice always a uniform probability model? Explain how it could or couldn’t affect Chris’ experimental probability.

15

3. Reasoning Abstractly … Students come up with their own experiments with uniform probability and perform 10, 20, 40 trials recording their results each time. Then students find the experimental probability at each stage and compare that to the theoretical probability. Describe how you think the experimental probabilities and theoretical probabilities would change (be similar or different) over time.

Hopes and Fears: Mini-Project: Step 1: Select two events that people might be afraid of or be hopeful for.

Fear: falling off a building OR getting bitten by a snake

Hope: Winning the lottery OR getting all A’s on your tests Step 2: Do some research to figure out the likelihood of either of the scenarios happening in a person’s life. (you will have to do some estimating!) Step 3: For each of the probability scenarios, explain how you came up with your theoretical probability. Step 4: Create a graphic representation of the two probabilities either by hand or by using technology (photoshop, Microsoft word, etc.). In it, show which is a more likely hope or a more likely fear. The idea is that you are MATHEMATICALLY PROVING to someone what they should be more hopeful for or fearful

of For cool ideas on how to represent ideas go to the website: http://visual.ly/

Step 5: Present your probability to your family at home and see if you can convince them to change their hopes and fears. You may also be asked to present this to your class.