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+ Unit 7: Probability Distributions Ms. Sanchez + Binomial Experiments Is a probability experiment that satisfies these conditions. There exist a fixed number of trials, where each trial is independent of the others. Only two outcomes: one is always given p = success p=1 – q q = failure q= 1 - p Central problem of a binomial experiment is to find the probability of x successes out of n trials. + Notation n: total number of trials p: probability p =1– q q: probability q of success in a single trial of failure in a single trial =1– p X: the random variables represent a count of the number of successes in n trials. [ie: possible success in the total trials] + Binomial Theorem The probability of exactly x successes in n trials is P(x) = (nCx)(p )(q x n-x ) + Example 5 contestants have a trail at the wheel of fortune. Say landing on gold is success and not landing on gold is failure. There is a 2% chance of landing on gold. What is the probability of exactly 3 successes? What is the probability of 3 or more successes? What is the probability that less than half of the contestants succeed? + E #1 Biologist are studying a new hybrid tomato. The probability of germinating is 0.70. The biologist plant 6 seeds and want to know the probability of 4 of them germinating What if they plant 8 seeds? + Graphing a Binomial Distribution A waiter knows that 7 out of 10 customers will leave a good tip when dining alone. During lunch hour he serves 5 people total. Find the Binomial Distribution of her lunch shift. 1. Probabilities of all possible outcomes, that the patrons will leave a good tip [0, 1, 2, 3, 4, 5] 2. Then construct a histogram. a. Place X values on the horizontal axis. b. Place P(x) values on the vertical axis. + E # 3. Find the probabilities of all outcomes, and create the histogram. 1. When Melecio practices penalty kicks, he makes 80% of them. For practice he kicks 8, draw the binomial distribution. 2. A satellite is powered by 5 solar cells. The probability that one cell won’t fail is 0.75. + Review of Normal Distribution + Binomial approximation to Normal Distribution. Mean: is the expected number of successes Standard m = np Deviation: s = npq + Example: Binomial to Normal When Cesar practices penalty kicks, he makes 80% of them. For practice he kicks 8, draw the binomial distribution. + Exercise #1: Draw the Binomial approximation to Normal 1. N = 100 p = 0.70 2. A satellite is powered by 10 solar cells. The probability that one cell won’t fail is 0.65. 3. Biologist are studying a new hybrid tomato. The probability of germinating is 0.45. The biologist plant 100 seeds + Geometric Distribution Geometric Distribution is a discrete probability distribution that satisfies these conditions. A trial is repeated until a success occurs. The trials are independent of each other P is the probability of success. X represents the number of trials in which the first success occurs. The probability that the 1st success will occur on trial x is x-1 P(x) = pq + Example: Geometric Distribution A patient is waiting for a suitable matching kidney donor. If the probability that a randomly selected donor is a suitable match is p=0.1. What is the probability that the patient will get a suitable donor after 3 attempts? +