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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science November 10, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 17 The Four-Step Method This is a good approach to questions of the form, What is the probability that ? Intuition will mislead you, but this formal approach gives the right answer every time. 1. Find the sample space. (Use a tree diagram.) 2. Define events of interest. (Mark leaves corresponding to these events.) 3. Determine outcome probabilities: (a) Assign edge probabilities. (b) Compute outcome probabilities. (Multiply along root-to-leaf paths.) 4. Compute event probabilities. (Sum the probabilities of all outcomes in the event.) Recitation 17 2 1 The Four-Door Deal Suppose that Lets Make a Deal is played according to different rules. Now there are four doors, with a prize hidden behind one of them. The contestant is allowed to pick a door. The host must then reveal a different door that has no prize behind it. The contestant is allowed to stay with his or her original door or to pick one of the other two that are still closed. If the contestant chooses the door concealing the prize in this second stage, then he or she wins. 1. Contestant Stu, a sanitation engineer from Trenton, New Jersey, stays with his original door. What is the probability that he wins the prize? The tree diagram is awkwardly large. This often happens; in fact, sometimes youll encounter infinite tree diagrams! Try to draw enough of the diagram so that you understand the structure of the remainder. Solution. Lets make the following assumptions: (a) The prize is equally likely to be behind each door. (b) The contestant is equally likely to pick each door initially, regardless of the prizes location. (c) The host is equally likely to reveal each door that does not conceal the prize and was not selected by the player. A partial tree diagram is shown below. The remaining subtrees are symmetric to the full-expanded subtree....
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