Overview

First students make a line that goes from “impossibility” (0) to “certainty” (1) and place where they believe events belong on the line. Then they play an extrasensory perception game.

Objective

Students will understand the definition of probability and use correct symbolism when writing the probability of an event.

Standards Addressed

Mathematics — Data Analysis

Grade 4

Probability, Benchmark F

09. Conduct simple probability experiments and draw conclusions from the results; e.g., rolling number cubes or drawing marbles from a bag.

10. Represent the likelihood of possible outcomes for chance situations; e.g., probability of selecting a red marble from a bag containing 3 red and 5 white marbles.

11. Relate the concepts of impossible and certain-to-happen events to the numerical values of 0 (impossible) and 1 (certain).

12. Place events in order of likelihood and use a diagram or appropriate language to compare the chance of each event occurring; e.g. impossible, unlikely, equal, likely, certain.

 

Grade 5

Probability, Benchmark H

07. List and explain all possible outcomes in a given situation.

Probability, Benchmark I

08. Identify the probability of events within a simple experiment, such as three chances out of eight.

09. Use 0, 1 and ratios between 0 and 1 to represent the probability of outcomes for an event, and associate the ratio with the likelihood of the outcome.

 

Grade 7

Probability, Benchmark K

08. Make predictions based on theoretical probabilities, design and conduct an experiment to test the predictions, compare actual results to predicted results, and explain differences.

Materials

• Multiple red, white and blue game tokens

• Paper bag

• Index cards

Procedure

1. Ask students to take out a piece of paper and draw a line about half-way down. Ask them to label the left side of the line as “Impossible” and the right side of the line as “Certain.”

2. Read each of the sentences below and ask them to put the letter of the sentence where they think it goes on the line.

You will grow another head.

You will take at least one more breath.

You will toss a penny and it will land heads-up.

You will go on a vacation next summer.

You will go to the moon sometime during your lifetime.

3. Have a discussion about the students’ placements. “A” should be considered impossible; “b” would be certain (although they sometimes argue that it’s not absolutely certain); “c” should be in the center because there’s an equally likely chance of getting either; and “d” and “e” are variable.

4. After this has been discussed, ask the students what percent of the time they could expect to toss a head on a penny. They should say 50 percent. Write 50 percent as a decimal and as a fraction and put 1⁄2 on the line above the letter “c.”

5. If this is one-half of the line, they should draw the conclusion that 1⁄2 is halfway between 0 and 1; therefore, an impossible event has a value of 0 and a certain event has a value of 1. All other events will have a value somewhere between 0 and 1 and can be written as a decimal, a fraction (ratio) or a percent.

6. Place five red, one white and three blue game tokens in a paper bag and record the quantities on the board.

7. Ask a student to choose a token and tell what he or she thinks the probability is for getting one of that color. Do this several times (replacing the token each time).

8. Ask the class to build a definition of probability from this activity. Record their ideas. Following is the correct definition:

   P(Event) =	# of observations favorable to the event
   	Total possible observations

9. Point out that the following equation is the correct way to write the probability of tossing a coin heads-up:

   P(heads) =	1		or P(heads) = .5 or P(heads) = 50%
   	2

    

10. Go back to you bag of tokens. This time, have the first student pull out a token and record the probability of the color drawn. Do not replace the token. Have a second student pull out a token and record the probability. Repeat this several more times. The students should see immediately that the total will change with each draw.

11. If the students need further help, use the following simulation: Put a number of tokens of different colors into the bag, only make each color equal to a letter grade. For example, red equals an A, blue equals a B, etc. Make sure you have the same number of tokens in the bag as there are number of people in the class. Tell the class that they are going to choose their report card grade by picking a token. Do this once with replacement and once without replacement. You don’t have to repeat the exercise many times — they catch on quickly.

12. Make sets of cards (one set for every two students) using index cards as follows:



13. Tell the students that we are going to test their extra-sensory perception (ESP) using probability.

14. Ask the students what the probability would be for picking each of these items by chance. The answer is P(shape) = 1⁄4 or .25.

15. Have the students pair up with a partner and each take out a piece of paper. On the paper each student should make a table like the following:

Correct	Incorrect

16. One partner mixes the cards up and concentrates on the symbol he or she is looking at. The other partner tries to guess what the card is and records if they answer correctly or incorrectly.

17. Allow about two minutes and instruct the students to change positions so the other partner is guessing.

18. Have the students calculate the percent they got correct.

19. By chance, they should have gotten 25 percent correct.

20. Students should compare the experimental probability (what actually happened) with the theoretical probability (what is supposed to happen) of .25.

    Students who guessed with greater than 25 percent accuracy have “good ESP.” Watch out! They could know what you’re thinking.

    Students who guessed with less than 25 percent accuracy probably shouldn’t go to Las Vegas today!

Evaluation

At the end of class, give the students the following scenario: “A student has five hats. Two are blue, two are red and one is white.”

Have them write the probability that they will pick a blue hat when choosing randomly. Check their responses and make sure that they have used correct symbolization.