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Chapter5_1 - Random Phenomena For random phenomena the...

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1 Chapter 5: Probability Read Chapter 5 Random Phenomena For random phenomena, the outcome is uncertain In the short-run, the proportion of times that something happens is highly random In the long-run, the proportion of times that something happens becomes very predictable Probability quantifies long-run randomness Law of Large Numbers As the number of trials increases, the proportion of occurrences of any given outcome approaches a particular number “in the long run” For example, as one tosses a coin, in the long run 1/2 of the observations will be a Head. Coin Flipping Applet Probability With random phenomena, the probability of a particular outcome is the proportion of times that the outcome would occur in a long run of observations Example: When rolling a die, the outcome of “6” has probability = 1/6. In other words, the proportion of times that a 6 would occur in a long run of observations is 1/6. Independent Trials Different trials of a random phenomenon are independent if the outcome of any one trial is not affected by the outcome of any other trial. Example: If you have 20 flips of a coin in a row that are “heads”, you are not “due” a “tail” - the probability of a tail on your next flip is still 1/2. The trial of flipping a coin is independent of previous flips. How do we calculate Probabilities? Calculate theoretical probabilities based on assumptions about the random phenomena . For example, it is often reasonable to assume that outcomes are equally likely such as when flipping a coin, or a rolling a die. Observe many trials of the random phenomenon and use the sample proportion of the number of times the outcome occurs as its probability. This is merely an estimate of the actual probability.
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2 Looking out for #1: The first digit (Benford s) law A phenomenon on the first digit of numbers. Numbers start with smaller digit rather than bigger digits In a logarithm book, Simon Newcomb noticed the pages started dirty and got cleaner toward the back Numbers more frequently begin with a ‘1’ than any other number, and the frequency decreases up to nine Scope of First Digit Law Census statistics Addresses Stock market Accounting figures Newspapers
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