## Explorations with Chance

- Lesson

In this lesson, students analyze the fairness of certain games by
examining the probabilities of the outcomes. The explorations provide
opportunities to predict results, play the games, and calculate
probabilities. Students should have had prior experiences with simple
probability investigations, including flipping coins, drawing items
from a set, and making tree diagrams. They should understand that the
probability of an event is the ratio of the number of successful
outcomes to the number of possible outcomes. This lesson was adapted
from "Activities: Explorations with Chance," which appeared in the
April 1992 issue of the *Mathematics Teacher**.*

Firsthand experiences with games of chance allow students the opportunity to develop intuitions and conjectures about the nature of probability. A more formal analysis follows from the students' direct experience of the game, providing both the experimental and theoretical probability.

The explorations will take 3-4 class periods to complete, depending on the students' backgrounds. To shorten the time, have groups complete different activities and then report their results to the class. It is important to allow enough time for students to discuss the strategies for approaching the tasks and the interpretations of the results.

Distribute the Is It Fair activity sheet. You may want to demonstrate the game and discuss question 1 together so as to prepare the students to address the question of fairness. Also, use this as an opportunity to review terms such as outcomes, trials, etc. Circulate and assess student developments.

Teachers may need to assist students in the construction of the tree diagram for the initial game, but groups should be encouraged to conduct their analyses of the variations with as little help as possible. The opportunity to create new versions of a chip game may generate some interesting ideas, and teachers should encourage students to present their creations to other groups or to the class. These sharings may lead to other concepts and questions.

Now distribute A Fair Hopper activity sheet. In this game, students have a chance to analyze the outcomes of tossing three coins. Once again, demonstrate the game, discuss question 1, and let students play. The rules for moving seem to indicate that player B will win (with 5 of the 7 squares), but surprisingly, these rules ensure that player A has a much greater chance of scoring. This counter-intuitive situation is likely to surprise students and helps them realize the importance of a theoretical analysis. The results of using four coins may also surprise students and reinforce these ideas.

Introduce the last game using the Happy Hopper activity sheet. Students must carefully trace the sequences of moves using the ordered-pair notation. Teachers may want to model this movement to the whole class. One tactic is to make the island with masking tape on the floor and have a student enact the sequence of moves (turns and jumps) by the happy hopper as others record what happens. Because it is very complex and the hopper can sometimes fall in the ocean, students may have to play the game many times to develop an adequate data base for their study. Students with some computer-programming background may be interested in developing a computer simulation of "happy hopper."

### Selected Solutions to Is It Fair Activity Sheet

**3. **Assume that the chips are r1-r2 and r1-b2. The outcomes are r1-r1, r1-b2, r2-r1, and r2-b2 (see **figure 1**). Thus, each player has the same chance (1/2) of scoring.

**5. **By adding a third chip, eight outcomes result: r1-r1-r1,
r1-r1-r2, r1-b2-r1, r1-b2-r2, r2-r1-r1, r2-r1-r2, r2-b2-r1, and
r2-b2-r2. Four outcomes are all red, so again each player could score
half the time.

**6. **The eight outcomes are r1-r1-r1, r1-r1-b2, r1-b2-r1,
r1-b2-b2, r2-r1-r1, r2-r1-b2, r2-b2-r1, and r2-b2-b2. The probability
that player A will score is 2/8, so the game is unfair.

**8. **The eight outcomes are shown in **fig. 2**. Only two
outcomes show three different colors, so player A has only a 2/8 chance
of scoring; player B has a 6/8 chance of scoring.

**9. **One possibility is to award player A three points each
time all the colors are different while still allowing player B one
point for tossing two identical colors. Another possibility is to let
stand player A's method of scoring and award player B a point only when
two of one color (white, blue, or red) land faceup.

*Selected Solutions to A Fair Hopper Activity Sheet *

**3. **The outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT (see **fig. 3**).
It is impossible to land on H or J. Only the outcomes of HHH and TTT
result in landing on K. Six outcomes result in landing on the I square;
thus the probability of player A scoring is 6/8.

**4. **One possibility is to award player B three points for each
chip on H, J, or K while limiting player A to one point for each chip
on I. Another is to alternate each turn so that, for instance, A gets I
the first time, then B gets I the second time, and so on.

**7. **Tosses of HHHH and TTTT result in landing on L (see **fig. 4**), but all other outcomes end on either J (8) or H (6).

**8. ** Two outcomes land on M, ten outcomes land on K, and twenty outcomes land on I.

### Selected Solutions to the Happy Hopper Activity Sheet

**2. **The hopper can escape only from square (3, 2); it can
never reach bridge (2, 4) because of an orientation change at (2, 3).
Similarly, it will never fall in the ocean at squares (4, 1), (4, 3),
(2, 0), (0, 1), or (0, 3).

Having arrived at the corner squares (1, 1), (1, 3), (3, 3), or (3,
1), half the time the hopper ends in the ocean and the other half the
time on squares (3, 2) or (1, 2) (see **fig. 6**). Having arrived
at (3, 2), half the time the hopper ends back at H, and half the time
it ends at the bridge (4, 2). Having arrived at (1, 2), half the time
it ends back at H and half the time in the ocean at (0, 2).

Thus, theoretically, after four sets of two tosses:

- 1/8 of the time the hopper is on bridge (4, 2)
- 1/8 of the time in the ocean at (3, 4)
- 1/8 of the time in the ocean at (1, 4)
- 1/8 of the time in the ocean at (1, 0)
- 1/8 of the time in the ocean at (3, 0)
- 1/8 of the time in the ocean at (0, 2)
- 1/4 of the time back at H

If after four sets of two tosses the hopper is at H, then it begins the circuit again. After the next four sets of two tosses, a theoretical probability of 1/32 exists of its ending at the bridge (4, 2), or in the ocean at each of squares (3, 4), (1, 4), (1, 0), (3, 0), or (0, 2). The probability of ending at H again is 1/16, and so on. Thus, the probability of ending up at the five positions in the ocean or the one position on the bridge approaches 1/6 as the probability of returning to H approaches zero. Thus, 5/6 of the time the hopper ends in the ocean; 1/6 of the time it ends on the bridge. The game is not fair.

**3. **Because the hopper can never reach (2, 0), adding another
bridge attached to (2, 1) does not improve the chances of escape. A
bridge at (0, 2) will give another possible escape route.

With two possible escape bridges, 1/3 of the time the hopper escapes
and 2/3 of the time it lands in the ocean: still not fair. For the game
to be fair, add a bridge at one of the spots at which the hopper falls
into the ocean (see **fig. 7**).

**6. **Five bridges could be added as shown in **figure 5**.

### References

- Hatfield, Larry L., "Activities: Explorations with Chance," The Mathematics Teacher, April 1992.
- National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics. Reston, Va.:The Council, 1989.

- Is It Fair Activity Sheet
- A Fair Hopper Activity Sheet
- Happy Hopper Activity Sheet
- Chips with faces of these colors: red-red, red-blue, red-white, and white-blue
- Cup
- Coins

### Learning Objectives

- Predict and determine if games are fair.
- Create a fair game from an unfair game.
- Organize and conduct a systematic investigation that includes empirical observations and theoretical analyses.

### NCTM Standards and Expectations

- Use simulations to construct empirical probability distributions.

- Compute and interpret the expected value of random variables in simple cases.

- Understand the concepts of conditional probability and independent events.

- Understand how to compute the probability of a compound event.