Slater Mazes

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Purpose

This is a level 5 statistics activity from the Figure It Out series.
A PDF of the student activity is included.

Achievement Objectives
S5-4: Calculate probabilities, using fractions, percentages, and ratios.
Student Activity

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Specific Learning Outcomes

calculate the probabilities of events occurring using fractions

Required Resource Materials
FIO, Level 4+, Statistics, Book Two, Slater Mazes, page 17
Activity

Activity One

This activity involves calculating the theoretical probability for the possible outcomes of an experiment. Although the slaters would not necessarily go down the various possible paths in the way suggested, the activity provides an interesting context for a probability investigation, especially as the mazes are, in effect, tree diagrams.
The activity reinforces the fact that some situations demand the ability to multiply and add fractions. However, if the students have a major problem with the fraction operations, it is possible to sidestep this by asking them to think in terms of a whole number of slaters (for example, 24) starting at the top and then tracking them through the various paths. Each time the slaters reach a junction where the path divides into two equally likely paths, half go down one and half go down the other. So if 24 approach a fork, 12 will go one way and 12 the other. If one of these groups of 12 approaches a second fork, 6 will go one way and 6 the other.
If they find this activity difficult, you could set it up as a game, using a dice and counter or a coin and counter to simulate the slater decision-making process. The students can play the game many times, keep details of the results, and collate them with the results of other students.
To do this, you will need copies of each maze on paper or card. The students place a counter at the starting point to represent a slater and then toss a coin. If the coin is a “head”, the slater chooses the left path; if a “tail”, the right path. The students toss the coin again each time the slater is faced with a choice. (Remind them that going back up the maze is not allowed.) When the slater reaches an exit, the students put a tally mark by that exit or on a table that has a column for each exit.
Suggest to the students that they play each game 24 times and record how many times the slater comes out of each exit. Ask them to examine their results and suggest, for the particular maze that they used, why they got the results they did. If there are others playing the game, suggest that they collate all their results so that a larger data set is obtained. The greater the number of results, the more likely it is that they will mirror the expected (theoretical) probabilities.

Answers to Activities

1. a. Exit B has a 1/2 chance because there are 2 paths the slater could follow to get there, whereas there is only 1 path to each of the others. (So exits A and C have a 1/4 chance.)
b. Proposals will vary. The best proposals will have a prize for just one exit, A or C; Jack won’t make much money if he rewards every player. The prize will need to cost less than 4 entries to the game, or Jack will be likely to make a loss, not a profit.
2. D and G have a 1/8 chance each. (There are 8 different paths to the exits, but only 1 of them ends up coming out at D and 1 at G.)
E and F have a 3/8 chance each. (3 of the 8 possible paths end up coming out at E and 3 at F.)
H and K have a 1/6 chance each. (There are 6 different paths to the exits, but only 1 of them ends up coming out at H and 1 at K.)
I and J have a 1/3 chance each. (2 of the 6 possible paths come out at each of these exits.)

tree diagram.
L has a 1/4 chance, M has a 1/2 chance (1/4 + 1/4 ), and N and O each have a 1/8 chance. This maze is different from the others in that one branch has an extra set of choices. (The tree diagram models the choices, with the extra set of choices shaded.)
3. P has a 1/2 chance, Q and R each have a 1/4 chance. The choices are modelled in this tree diagram:

tree diagram.
S has 1/2 a chance, T has a 1/4 chance, and U and V each have an 1/8 chance. The choices are modelled in this tree diagram:
 

tree diagram.

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Level Five