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DISCRETE PROBABILITY DISTRIBUTIONS Farrokh Alemi Ph.D. Discrete Probability Distributions Bernoulli Geometric Binomial Poisson Definitions Density function Definitions Cumulative probability function Definitions Expected Value Expected Value Expected Value Expected Value Example Example Example Example Probability density Events function 0 medication errors 0.74 1 medication error 2 medication errors Otherwise Cumulative distribution function 0.74 0.22 0.04 0.96 1 0 1 Example Example Density & Cumulative Distributions Typical Probability Density Functions Bernoulli Geometric Binomial Poisson Bernoulli Probability Density Function Mutually exclusive Bernoulli Probability Density Function Exhaustive Bernoulli Probability Density Function Events Does not occur Occurs Probability density function 1-P p Bernoulli Probability Density Function Independent Bernoulli Trials Independent Bernoulli Trials Geometric Density Function Geometric Density Function Geometric Density Function Do One No medication errors have occurred in the past 90 days. What is the maximum daily probability of medication error in our facility? Do One The time between patient falls was calculated to be 3 days, 60 days and 15 days. What is the daily probability of patient falls? Binomial Probability Distribution Number of k occurrences of the event in n independent trials Independent Bernoulli Trials Independent Bernoulli Trials Independent Bernoulli Trials Independent Bernoulli Trials P x (1-P) x P Binominal Probability Distribution Different combinations Success probabilities Failure probabilities Binomial Probability Distribution Binomial Probability Distribution Binomial Probability Distribution 6 Trials of Binomial p=1/2 6 Trials of Binomial p=1/2 6 Trials of Binomial p=0.05 Example If the monthly probability of elopement is 0.05, how many patients will elope in 2 years? Example If the monthly probability of elopement is 0.05, how many patients will elope in 2 years? Example If the daily probability of death due to injury from a ventilation machine is 0.002, what is the probability of having 1 or more deaths in 30 days? Example If the daily probability of death due to injury from a ventilation machine is 0.002, what is the probability of having 1 or more deaths in 30 days? Do One Which is more likely, 2 patients failing to comply with medication orders in 15 days or 3 patients failing to comply with medication orders in 30 days. Poisson Density Function Large number of trials Small probabilities of occurrence Poisson Density Function k Poisson Density Function k Take Home Lesson Repeated independent Bernoulli trials is the foundation of many distributions Do One What is the probability of observing one or more security violations, when the daily probability of violations is .01 and we are monitoring the organization for 4 months. Do Another How many visits will it take to have at least one medication error if the estimated probability of medication error in a visit is 0.03?