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Course 1, Unit 8 - Patterns in Chance

The Patterns in Chance unit introduces students to sample spaces, probability distributions, the Addition Rule, simulation, and geometric probability. Important probabilistic concepts explored include mutually exclusive events and the Law of Large Numbers.

In Lesson 1, students learn to use sample spaces and probability distributions to calculate probabilities. In Lesson 2, students learn how to estimate probabilities using simulation and random number generators and how to calculate probabilities using geometric (area) models.

Simulation is the modeling of a probabilistic situation using random devices such as coins, spinners, and random digits. Simulation is useful throughout instruction on probability. Setting up a simulation helps students clarify their assumptions about such things as whether trials are independent. Simulation helps develop students' intuition about probabilistic events. And, perhaps most importantly, students who have been introduced to simulation have a feeling of control over probability. They know that they can estimate the answer to any probability problem that arises.

Key Ideas from Course 1, Unit 8

  • Probability distribution: a description of all possible numerical outcomes of a random situation, along with the probability that each occurs. The description may be in table, formula, or graphical form. (See page 534.)

  • Mutually-exclusive events: Two events are said to be mutually exclusive (or disjoint) if it is impossible for both of them to occur on the same outcome of a probability experiment. (See pages 537-539 for the development of this idea.)

  • Addition Rule: The rule for computing the probability that event A or event B occur is developed on page 540.

  • Simulation model: a way of modeling a real situation that involves randomly occurring events. For example, suppose that 45% of a population has type O blood. Here's how to use simulation to estimate the probability that 4 random people entering a blood donor bank have blood type O. Model the selection of one person chosen at random from the population by assigning the numbers 1-45 to represent people with type O blood and 46-100 to represent other people. Choosing 4 numbers at random from 1 to 100 and counting how many are 1-45 would simulate 4 random people entering a blood donor clinic and being identified by blood type. Simulations can be run using the CPMP-Tools public domain suite of software tools. (See pages 553-555.)

  • Law of Large Numbers: The Law of Large Numbers says that the more runs there are in a simulation, the better your estimate of the probability tends to be. (See page 555.)

  • Geometric probability: Area models can be used to model chance situations where the numbers are selected at random from a continuous interval. (For an example, see page 568 Problem 1.)

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