Background: During the 1500’s Cardano was one of the first people to study probability (probably because he was a noted gambler). In the 1600’s Fermat in his correspondence with Pascal develop the theory of probability. Pascal also worked on the concept of mathematical expectation and there is a distribution in probability named after him. In the 1700’s DeMoivre published the Doctrine of Chances, which contained a series of solved problems, such as: “Suppose that three tickets will be given prizes in a lottery having 40,000 tickets. What is the chance of winning at least one prize if you buy 8000 of those tickets?”

Students often hear and occasionally use statements of probability in their daily lives. They note the weather forecasts when they wonder whether a game will be held or school canceled. They also use more ambiguous general phrases such as not likely, no way, and probably but, as with much of everyday speech, there are many misuses of probability terminology and concepts.

Purpose: To introduce and develop the concept of probability.

Materials: Sack; marbles of two different colors - 100 of one color (blue), 25 of another color (green).

Procedure:

1. Announce to the class: “Let’s do an experiment. We will try to find out - without looking in the sack and counting - whether there are more blue or more green marbles in the sack.”
2. Have four students draw five marbles each from the sack. (Make sure that the marbles are put back into the sack after each draw.)
3. Have every student record the numbers and colors of marbles for each of the four draws.

1. On the basis of the first four draws how many marbles of each color are there in the sack? D. Let each student in the rest of the class draw five marbles each from the sack. (Be sure to put the marbles back in the sack after each drawing.)
2. What are the totals for each color of marble?
3. Do you think there were more marbles of one color than the other? Why?
4. If so, what do you think the ratio of one color to the other might be? G. Open the sack and count the number of marbles of each color.
5. What is the ratio of one color to the other color?

The purpose of this activity is to introduce basic information on probability and statistics. It can be used as an introduction to a unit on probability. It should be followed up with a discussion about how probability is used in the real world. This activity can be made as simple or as complex as necessary depending on grade level.

Materials:

• Two sets of hands
• Paper
• Pencil

Procedures:

Divide the class into pairs and have them play the game eighteen times. A rock is a closed fist. Paper is palm on palm, and scissors is the number two horizontally. The student hits their other hand twice, and on the third time gives the symbol they wish. A rock beats scissors. Paper beats rocks, and scissors beats paper. Instruct the students to keep a record of wins and losses.

Once the class has finished, record the results for player A is one color, and player B in another color. Then, the students can figure mean, mode, and range each set of data.

Now draw a tree diagram to show all possible outcomes.

Answer the following questions to determine if the game is fair.

1. How many outcomes does the game have? (9)
2. Label each possible outcome on the tree diagram as to win for a, b, or tie.
3. Count the number of wins for A. (3)
4. Find the probability A will win in any round. (3/9 = 1/3)
5. Count the number of wins for B. (3)
6. Find the probability B will win in any round. (3/9 = 1/3)
7. Is the game fair? Do both players have an equal probability of winning any round? (yes)
8. Compare the mathematical model with what happened when the students played the game.
9. How do you think probability is used in the real world? See how many areas you can list that use probability.

Historical Note:

John Graunt (1620-1674) created the London Life Table. He found 100 people and charted the number of survivors after 0 years, 6 years, and so on. It is very interesting to see that after only 6 years, 36 people had already died.