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PROBABILITY The probability PA of any event A satisfied 0PA1If S is the sample space in a probability model the PS1 Two events A and B are disjoint if they have no outcomes in common and so can never occur together If A and B are disjoint PA or BPAPB cThe complement of any event A is the event that A does not occur PA1PA If a random phenomenon has k possible outcomes all equally likely then each individual outcome has probability 1k The probability of any event A is PAcount of outcomes in Acount of outcomes in Scount of outcomes in Ak Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs If A and B are independent PA and BPAPB A discrete random variable X has a finite number of possible values The probability distribution of X lists the values and their probabilities The probabilities must satisfy two requirements every probability is a number between 0 and 1 and the sum of all probabilities must add up to 1 Standardizing normal random variables ZXuo is a standard normal random variable having the distribution N01 Addition rule PA or BPABPAPBPA and B Union the probability that event A or event B occurs at least one occurs Multiplication rule PA and BPABPAPB Intersect the probability that events A and B both occur the intersection of any collection of events is the event that all of the events occur Tree diagrams The multiplication rule says that the probability of reaching the end of any complete branch is the product of the probabilities written on its segments The probability of any outcome is found by adding the probabilities of all branches that are part of that event SAMPLING DISTRIBUTIONS Facts about sample means 1 Sample means are less variable than individual observations 2 Sample means are more normal than individual observations Mean of a sample mean uxu u is the population mean Standard deviation of a sample mean oxosqrtn o is the population sd n is the sample size Standard deviation is 2calculated by sqrt1n1xix where xithe individual entry xmean of all entries n1sample size1Sampling distribution of a sample mean If a population has the Nuo distribution then the sample mean ux of n independent observations has the Nu osqrtn distribution Central Limit Theorem when n sample size is sufficiently large the sampling distribution of the sample mean ux is approximately normal UxNu osqrtn Rules for means If x and y are random variables then uxyuxuy 2222The variance of x is o x1uxp1x2uxp2xnuxpnxThe standard deviation of x is the square root of the variance 222222Rules for variance If x and y are independent random variables then o o o o box y x y a bx x
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