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Section 6.3 Geometric Distribution Statistics In the last section we looked at the probability that a certain number of people in a group will have a certain characteristic. For example, in a group of 15 Congolese citizens picked at random, we could determine that exactly 3 of them would live in Kinshasa. Geometric distributions look at a different type of probability. An example of a geometric distribution type of probability would be if I were to interview Congolese citizens picked at random, what is the probability that the third person would be the first person I interviewed from Kinshasa? Geometric distribution problems are sometimes called ____________________ problems because you are waiting until the first event of something happens. These problems also differ from binomial distribution problems because the number of ______________ is not fixed. The number of trials for each situation is the number of times it takes until the first occurrence of the event of interest. What reasoning can we use to determine the probability for our first “success” to be a certain trial number? One condition that is the same as for binomial distributions is that we have two possible outcomes, ___________________ or __________________. If the probability of success is _______, the probability of failure will be ____________ (let us call that q for now). Using p and q as probabilities for “success” or “failure”, then, the probability for our first success being on the fourth trial, say, would be: P X 4 _______ _______ _______ _______ As with the binomial distribution, there are four conditions that must be met for a geometric distribution: 1.) They are ________________. (Each trial will have either a _________ or a __________________. 2.) Each trial is __________________ of the others (the result of one trial has no ______________ on another trial). 3.) The trials continue until the first _______________ occurs. 4.) The probability, ______, of success is the ____________ for each trial, ______ < ______ < ______. The distribution of the random variable X that counts the number of trials needed until the first “success” occurs is called a ________________________. The probability that the first success occurs on the X = ________ trial is: ______________________________________________ for k = 1, 2, 3, . . . The formulas for the Expected number of trials before the first success and the standard deviation of the expected number are: ______________________ and _____________________ If you are interested in the number of trials before the nth success occurs, you just multiply n by the expected value: Expected number of trials before the nth success = __________________ The TI calculators have a built in function for computing binomial probabilities. The first is geometpdf(p, k). It is found by doing 2nd VARS, under DIST go down to D. Typing in p and k and ENTER gives the probability that the first occurrence will be on the kth trial. If k is entered in { brackets (2nd”(“ and 2nd “)”) you can enter any number of k values separated by “,” and get probability returns for all of those k values. The second is geometcdf(p,k). This is found by doing 2nd VARS, under DIST go down to E. Typing in p and k and ENTER gives the cumulative probability that the first occurrence will be on the 1st, 2nd, 3rd, ... ,to kth trial.