Illuminations: Stick or Switch?

# Stick or Switch?

 This lesson plan presents a classic game-show scenario. A student picks one of three doors in the hopes of winning the prize. The host, who knows the door behind which the prize is hidden, opens one of the two remaining doors. When no prize is revealed, the host asks if the student wishes to "stick or switch." Which choice gives you the best chance to win? The approach in this activity runs from guesses to experiments to computer simulations to theoretical models. This lesson was adapted from an article written by J. Michael Shaughnessy and Thomas Dick, which appeared in the April 1991 issue of the Mathematics Teacher.

### Learning Objectives

 Students will: use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations compute probabilities for simple compound events, using such methods as tree diagrams and area models

### Materials

 Spinner with three equal areas, such as the Adjustable Spinner Simple Monty Hall Simulation Generalized Monty Hall Simulation

### Extensions

 To illustrate the "Monty phenomenon" one more time, consider this situation. Suppose that you enter a sweepstakes and subsequently receive an announcement that after a random drawing out of a million entries, the winning ticket number is one of six listed numbers and your entry is among the six. What is the probability that you hold the winning ticket? Many people have actually had this experience and realize that their entry still has only a one-in-a-million chance of being a winner. However, if a new random drawing were made from only these six listed entries, then the situation would be entirely different. Your chance of winning would greatly improve. However, simply listing your entry alongside five others "after the fact" does not magically improve your chances to one in six, although that perception is hoped for by those running such sweepstakes.

### NCTM Standards and Expectations

 Data Analysis & Probability 6-8Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations. Data Analysis & Probability 9-12Understand how to compute the probability of a compound event. Understand the concepts of conditional probability and independent events

2 periods

### NCTM Resources

 More and Better Mathematics for All Students
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