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Discrete Probability Distributions Probability Distributions A random variable x represents a numerical value associated with each outcome of a probability experiment. It is DISCRETE if it has a finite number of possible outcomes. It is CONTINUOUS if it has an uncountable number of possible outcomes (represented by an interval) The number of books in a university library. The amount of snow (in inches) that fell in Nome, Alaska last winter. A list of each possible value and its probability. Must satisfy 2 conditions: 1. 2. 0 < P(x) < 1 ΣP(x) = 1 The # of games played in the World Series from 1903 to 2009 # of games 4 5 6 7 played Frequency 20 23 23 36 8 3 The # of 911 calls received per hour. X 0 P(x) 0.1 1 2 3 4 5 6 7 0.10 0.26 0.33 0.18 0.06 0.03 0.03 Notation: E(x) Expected value represents what you would expect to happen over thousands of trials. SAME as the MEAN!!! E(x) = µ = Σ[x·P(x)] If x is the net gain to a player in a game of chance, then E(X) is usually negative. This value gives the average amount per game the player can expect to lose. A charity organization is selling $5 raffle tickets. First prize is a trip to Mexico valued at $3450, second prize is a spa package valued at $750. The remaining 20 prizes are $25 gas cards. The number of tickets sold is 6000. Binomial Distributions CONDITIONS: 1. there are a fixed number of independent trials (n = # of trials) 2. Two possible outcomes for each trial, Success or Failure. 3. Probability of Success is the same for each trial. p = P(Success) and q = P(Failure) 4. random variable x = # of successful trials If binomial, ID ‘success’, find n, p, q; list possible values of x. If not binomial, explain why. From past records, a clothing store finds that 26% of people who enter the store will make a purchase. During a onehour period, 18 people enter the store. The random variable represents the # of people who do NOT make a purchase. To find the probability of exactly x number of successful trials: P(x) = nC x · x p · n –x q A surgical technique is performed on 7 patients. You are told there is a 70% chance of success. Find the probability that the surgery is successful for A) exactly 5 patients B) at least 5 patients C) less than 5 patients Construct a probability distribution, then find mean, variance, and standard deviation for the following: One in four adults claims to have no trouble sleeping at night. You randomly select 5 adults and ask them if they have trouble sleeping at night. Consider “success” to mean no trouble sleeping.