STA6166 F06-8 Probability Continued

STA6166 F06-8 Probability Continued - Chapter 4: Continued...

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Chapter 4: Continued Some Definitions (revisited) Random phenomenon : Something whose outcome is uncertain: flipping a coin, tossing a die, a baseball game, selecting a person randomly from the class and recording the person’s height in inches. Outcome : one possible result of a random phenomenon (e.g., “heads” and “tails” are the two possible outcomes of a coin flip; possible outcomes for measuring a random person’s height in inches are numbers, almost certainly between 0 and 100) Event : a collection of outcomes (e.g., getting an even number on the roll of a die; this event is a collection of three outcomes: 2, 4, and 6). Trial : a single realization of a random phenomenon (e.g., a single coin flip or die roll) Independent trials : trials where the outcome of one trial doesn’t affect the outcome of any other trial Probability of an event : long-run relative frequency in independent trials of a random phenomenon Law of Large Numbers : The long-run relative frequency of an event in repeated independent trials gets closer and closer to the true relative frequency (the true probability) as the number of trails increases. Simulation of a large number of coin tosses: Trial (log scale) Cumulative proportion 1 10 100 1000 10000 0.0 0.2 0.4 0.6 0.8 1.0 10000 trials Trial (log scale) 1 10 100 1000 10000 10 simulations, 10,000 trials each Note: a simulation model for a random phenomenon is just that: a model for the random phenomenon, not necessarily an exact representation of reality. In reality, the probability of a heads may not be exactly .5 and may depend on the coin and who flips it. In addition, the trials may not be independent (why might they not be?) But the model is generally a very good approximation to the reality of coin flipping. There is no “Law of Averages”!! One misinterpretation of the Law of Large Numbers is the so- called “Law of Averages.” Essentially, this “law” states that, in independent trials, if an outcome has not occurred in recent trials then it has a higher chance of occurring in subsequent trials. This is based on the belief that this must happen in order to make the Law of Large Numbers work – that if the
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2 cumulative relative frequency of an event has deviated from what is expected then the process must “make up” for this in order to make the long-run relative frequency equal to what is expected. This belief is false as illustrated below. Suppose that, just by chance, we obtain 100 heads on the first 100 flips of a fair coin.
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This note was uploaded on 11/23/2011 for the course STA 6166 taught by Professor Staff during the Summer '08 term at University of Florida.

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STA6166 F06-8 Probability Continued - Chapter 4: Continued...

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