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prelim 2 notes part 2

prelim 2 notes part 2 - Probability The proportion of the...

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Probability- The proportion of the time an event would occur in a very long sequence of repetitions. Any probability is a number from zero up to one that describes the long run relative frequency of a particular outcome. A probability of zero connotes impossibility, while a probability of one indicates certainty. Sum of all probabilities is one . For any event A, 0 < P(A) < 1 1. Probabilities are between 0 and 1 (as proportions) 2. P(event does not occur) = 1 – P(event does occur) Complement Rule – The probability of an event occurring is 1 minus the probability that it doesn’t occur P(A) = 1- P(A C ) A C is called a Complement of A, where the set of outcomes are NOT in the event A. Sample Space- The collection of all possible outcomes. The probabilities of the outcomes must sum to one. Proportion- A value between 0 and 1 (or equivalently between 0% and 100%) that indicates the fraction of individuals in some group that satisfy some condition. The sample proportion of successes in many outcomes will settle down to the true proportions. Were this not so, we would learn little about the world from observed data. However, the sample proportion is guaranteed to be close to the true proportion ONLY in the long run. The Law of Large Numbers – The long-run average of random outcomes settles down to the mean of the distribution of the values of the outcomes. OR that the sample averages of random quantities will approach the true population mean as the sample size increases. The average of Randomly Generated quantities approaches the true mean of the distribution of possible outcomes. Event – Collection of possible outcomes. The probability of an event is the SUM of the probabilities of its outcomes. PROBABILITY RULES Addition Rule for Disjoint Events- If Events A and B are Disjoint, THEN P(A or B) = P(A) + P(B) - For two disjoint events A and B, the probability that one or the other occurs in the sum of the probabilities of the two events - Disjoint (or Mutually Exclusive) – Events have no outcomes in common Relative Frequencies Number of Outcome A / Total # Probability (A) General Addition Rule – Add the probabilities of two events and then subtract out the probability of their intersection. This does not require disjoint events. P(A or B) = P(A) +P(B) – P(A and B) For any two events, A and B, the probability of A or B is P(A or B) = P(A) + P(B) – P(A and B) Conditional Probability –It takes into a account a given condition. To find P(B|A), or the probability of B given A, restrict attention to the outcomes in A. Then find what fraction of those outcomes B also occurred. P(B|A) = P(A and B) / P(A) General Multiplication Rule – Rule for compound events that doesn’t require the events to be independen t. P(A and B) = P(A) x P(B|A) -
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