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4.1-4.2 - Probability Randomness;Probabilitymodels...

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    Probability Randomness; Probability models
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Randomness and probability Probability models: sample spaces, events Assigning probabilities: finite number of outcomes Basic probability rules
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Random phenomena vs deterministic phenomena
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A phenomenon is random if individual outcomes are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions. Randomness and probability The probability of any outcome of a random phenomenon can be interpreted as the proportion of times the outcome would occur in a very long series of repetitions ( long-term relative frequency ).
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Coin toss The result of any single coin toss is random. But the result over many tosses is predictable, as long as the trials are independent (i.e., the outcome of a new coin flip is not influenced by the result of the previous flip). First series of tosses Second series The probability of heads is 0.5: approximate proportion of times you get heads in many repeated trials.
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A probability model describes mathematically the outcome of a random phenomenon. It consists of two parts: 1) S = Sample Space : This is a set, or list, of all possible outcomes of a random phenomenon. 2) A way of assigning probability to each possible outcome. Probability models Example: Probability Model for a Coin Toss : S = {Head, Tail} Probability of heads = 0.5 Probability of tails = 0.5
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Probabilities range from 0 to 1. Because some outcome must occur in every trial, the sum of the probabilities of all possible outcomes must be exactly 1. If we have a biased coin, say P (head) = 0.3, then P (tail) = 0.7 If P (head) = 2 P (tail), then P (head) = 2/3
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We can assign probabilities either: empirically from our knowledge of numerous similar past events Mendel discovered the probabilities of inheritance of a given trait from experiments on peas without knowing about genes or DNA.
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