l17_prob_intro - 6.042/18.062J Mathematics for Computer...

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6.042/18.062J Mathematics for Computer Science April 14, 2005 Srini Devadas and Eric Lehman Lecture Notes Introduction to Probability Probability is the last topic in this course and perhaps the most important. Many algorithms rely on randomization. Investigating their correctness and performance re- quires probability theory. Moreover, many aspects of computer systems, such as memory management, branch prediction, packet routing, and load balancing are designed around probabilistic assumptions and analyses. Probability also comes up in information theory, cryptography, artificial intelligence, and game theory. Beyond these engineering applica- tions, an understanding of probability gives insight into many everyday issues, such as polling, DNA testing, risk assessment, investing, and gambling. So probability is good stuff. 1 Monty Hall In the September 9, 1990 issue of Parade magazine, the columnist Marilyn vos Savant responded to this letter: Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what’s behind the doors, opens another door, say number 3, which has a goat. He says to you, ”Do you want to pick door number 2?” Is it to your advantage to switch your choice of doors? Craig. F. Whitaker Columbia, MD The letter roughly describes a situation faced by contestants on the 1970’s game show Let’s Make a Deal , hosted by Monty Hall and Carol Merrill. Marilyn replied that the con- testant should indeed switch. But she soon received a torrent of letters— many from mathematicians— telling her that she was wrong. The problem generated thousands of hours of heated debate. Yet this is is an elementary problem with an elementary solution. Why was there so much dispute? Apparently, most people believe they have an intuitive grasp of probability. (This is in stark contrast to other branches of mathematics; few people believe they have an intuitive ability to compute integrals or factor large integers!) Unfortunately, approxi- mately 100% of those people are wrong . In fact, everyone who has studied probability at
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2 Introduction to Probability length can name a half-dozen problems in which their intuition led them astray— often embarassingly so. The way to avoid errors is to distrust informal arguments and rely instead on a rigor- ous, systematic approach. In short: intuition bad , formalism good . If you insist on relying on intuition, then there are lots of compelling financial deals we’d love to offer you! 1.1 The Four-Step Method Every probability problem involves some sort of randomized experiment, process, or game. And each such problem involves two distinct challenges: 1. How do we model the situation mathematically?
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