• Coins and square grid (the diameter of the coin should be less than the length of a square on the grid, possibilities are plastic lids on floor tiles, or coins on graph paper). Note that a blank grid for coins is provided on the last page of this lesson.
• Pencil and paper for record keeping and note taking
• Calculator
DEVELOPMENT OF THE LESSON
A. Introduction / Motivation: Recall How to Assign Probabilities to Events Begin the session with a recall of the notion of the PROBABILITY (or Chance) of events, in the context of random processes where possible outcomes can be determined beforehand, but not whether an outcome will occur.
Mention to students that probabilities of events may be assigned:
(a) theoretically by assuming understanding of situations in the events, such as symmetry or equal-likely outcomes (e.g. fair coin, fair dice being tossed so
outcomes are equally likely), or if the events are related to areas of geometric objects (this is called “geometric probability”)
(b) subjectively with personal assessment of the situation (e.g. when a student tells his friend that he has 50 percent chance of passing the quiz, or the
probability that a student can swim around the world in 24 hours is zero (0)) (c) empirically by collecting data from repeated trials or experiences, and getting the proportion of times an event occurs (e.g. observing 10 patients, noticing that 6 of them responded to a medicine within one hour of the treatment, and thus, stating that the probability of response within an hour of receiving the treatment is 60 percent)
Inform students that a few hundred years ago, people enjoyed betting on coins tossed onto the floor ... Would they cross the line of the grid or not? Georges-Louis Leclerc, Comte de Buffon (1707 – 1788), a French mathematician, started thinking about this problem more systematically, expressing it as follows:
“What is the probability that a coin, tossed randomly at a grid, will land entirely within a tile rather than beyond the tile boundaries? (For the purposes of this
lesson, we will assume that the diameter of the coin is less than the length of a side of the tile.)
The ‘Buffon coin problem’ is an exercise in geometric probability, where probabilities are viewed as the proportions of areas (lengths or volumes) of
geometric objects under specified conditions. Examples of questions that deal with geometric probabilities are:
What is the probability of hitting the bull’s eye when a dart is thrown randomly at a target, given the target has a diameter of 24 cm, and the bull’s eye has a diameter of 10 cm?
What is the probability that a four-colored spinner, with a diameter of 20 cm, will land on red?
Geometric probabilities can be estimated using empirical approaches, or identified exactly using analytical methods (theoretical probability).
An empirical probability is the proportion of times that an event of interest occurs in a set number of repetitions of an experiment.
Example: Throw 100 darts at the target.
15 darts hit the bull’s eye.
The empirical probability of hitting the bull’s eye is 15/100 = 3/20.
Spin the spinner 50 times.
Spinner lands on red 12 times.
Empirical probability = 12/50 = 6/25.
A theoretical probability is the proportion of times an event of interest is expected to occur in an infinite number of repetitions of an experiment. For a geometric
probability, this is the ratio of the area of interest (e.g. bull’s eye) to the total area (e.g.
target).
Ask learners how they can identify an empirical solution to this Buffon’s coin problem?
(This corresponds to prompt 1 on the task sheet.)
C. Investigation on Empirical Probability: Buffon’s Coin Problem I. Problem Formulation: Buffon’s coin
What is the probability that a coin, tossed randomly at a grid, will land entirely within the tile rather than beyond the tile boundaries? (Recall that in this activity, we assume that the diameter of the coin is less than the length of a side of the tile.) II. Design and Implement a Plan to Collect the Data
• Discuss this as a class: How would you identify an empirical solution to Buffon’s coin problem? (See Item 1 on Activity Sheet.)
Area of bull’s eye = Area of target =
Theoretical probability of hitting bull’s eye =
Area of red section = Area of spinner =
Theoretical probability of landing on red =
After learners propose tossing coins at a grid, discuss details of the
experiment. Divide the class into groups of five students. How many times will each group throw the coin? How will the coin be tossed? Will they count the times the coin lands on a boundary or the times it lands entirely within the tile? Will each group do this the same way? What difference will it make if they do not? (For purposes of later discussion, it will be helpful if everyone considers the event that the coin lands entirely within a tile.) Who will record the outcome of each toss? How will this count translate into an empirical probability?
• Experiment: Instruct each group to conduct the experiment, as designed by the class. (See Item 2 of Activity Sheet.)
III. Analyze the Data
Instruct each group to use the data they gathered to compute the empirical probability of the event they considered.
IV. Interpret the Results
• Discuss as a class:
o Summarize the empirical probabilities generated by the groups on the blackboard. Ask learners what they observe about the empirical
probabilities computed by the groups. (They are not all the same, many may be similar, a few may differ by a lot, if the experiment were repeated different answers would be obtained).
o Is it possible to get a more stable answer? (Yes, repeat the experiment more times, combine data from different groups).
o Ask students what they would expect to see if the coin could be tossed an infinite number of times. Why would they expect to see this? (See Item 3 of Activity Sheet.)
D. Investigation on Geometric Probability I. Problem Formulation: Buffon’s coin
Recall Buffon’s coin problem: What is the probability that the coin, tossed randomly at a grid, will land entirely within a tile rather than beyond the tile boundaries?
II. Solution to the Problem
• Discuss this as a class: How will they identify a theoretical solution to Buffon’s coin problem? (See Item 4 of Activity Sheet.)
Outline the process here: Identify the shape of the region within the tile where the coin must land to be entirely within the tile. Look at the ratio of the area of that shape to the area of a tile. Learners must work out the details with group mates in the next segment.
Answer: The Probability of a Crack Crossing
Our main interest is in the event C that the coin crosses the tiles. However, it turns out to be easier to describe the complementary event Cc that the coin does not cross a tile.
If the tile has unit length, and radius r < ½, then
P(Cc )= (1 - 2r)2, Thus, P(C) = 1 - (1 - 2r)2
• Explore: Again working in groups, ask students:
o To formulate a conjecture about the relationship between theoretical and empirical probabilities. (See Item 5 of Activity Sheet.) o To identify the shape of the region within the tile in which the coin
must land to be entirely within a tile. (This will be challenging for some learners. The key is to consider where the center of the coin lands and how close the center can be to the edge of a tile while the coin is not on the boundary.)
III. Analyze the Data
Instruct each group to use observations from the experiment, and the group discussion to compute for the theoretical probability that a coin tossed randomly at a grid lands entirely within a single tile. (See Item 6 of Activity Sheet.)
IV. Interpret the Results
• Discuss as a class: Summarize learners’ answers on the board. Discuss observations about these solutions. Bring the class to a consensus on the solution. Observe that there is only one solution and it will not vary with further investigation.
• Synthesis
What seems to be the relationship between empirical and theoretical probabilities? (See Item 7 of Activity Sheet.)
KEY POINTS
• Empirical probabilities (obtained from observing the proportion of times an event occurs in repeated trials) may differ, but the long run frequency of empirical probabilities will stabilize toward the theoretical probability. (As the number of trials increases, the empirical probability tends to converge to the theoretical one).
• For some situations, we can calculate the theoretical probabilities as geometric probabilities, when events pertain to areas of geometric objects.
• Sometimes, we associate probabilities subjectively, according to personal assessment of the likelihood of an event to occur.
ACTIVITY SHEET 2-02