1 Probability Distributions for Discrete Variables Farrokh Alemi Ph.D. Professor of Health Administration and Policy College of Health and Human Services, George Mason University 4400 University Drive, Fairfax, Virginia

2 Lecture Outline 1. What is probability? 2. Discrete Probability Distributions 3. Assessment of rare probabilities 4. Conditional independence 5. Causal modeling 6. Case based learning 7. Validation of risk models 8. Examples

3 Lecture Outline 1. What is probability? 2. Discrete Probability Distributions Bernoulli Bernoulli Geometric Geometric Binomial Binomial Poisson Poisson 3. Assessment of rare probabilities 4. Conditional independence 5. Causal modeling 6. Case based learning 7. Validation of risk models 8. Examples

4 Definitions Function Function Density function Density function Distribution function Distribution function

5 Definitions Events Probability density function Cumulative distribution function 0 medication errors medication error medication errors0.041 Otherwise01

6 Expected Value Probability density function can be used to calculate expected value for an uncertain event. Probability density function can be used to calculate expected value for an uncertain event. Expected Value for variable X Probability of event “i” Value of event “i” Summed over all possible events

7 Calculation of Expected Value from Density Function Events Probability density function Value times probability 0 medication errors0.900*(0.90)=0 1 medication error medication errors0.04 Otherwise0

8 Calculation of Expected Value from Density Function Events Probability density function Value times probability 0 medication errors0.900*(0.90)=0 1 medication error medication errors Otherwise00

9 Calculation of Expected Value from Density Function Events Probability density function Value times probability 0 medication errors medication error medication errors Otherwise00 Total0.12 Expected medication errors

10 Exercise Chart the density and distribution functions of the following data for patients with specific number of medication errors & calculate expected number of medication errors Chart the density and distribution functions of the following data for patients with specific number of medication errors & calculate expected number of medication errors

11 Probability Density & Cumulative Distribution Functions

12 Exercise If the chances of medication errors among our patients is 1 in 250, how many medication errors will occur over 7500 patients? Show the density and cumulative probability functions. If the chances of medication errors among our patients is 1 in 250, how many medication errors will occur over 7500 patients? Show the density and cumulative probability functions.

13 Typical Probability Density Functions Bernoulli Bernoulli Binomial Binomial Geometric Geometric Poisson Poisson

14 Bernoulli Probability Density Function Mutually exclusive Mutually exclusive Exhaustive Exhaustive Occurs with probability of p Occurs with probability of p

15 Exercise If a nursing home takes care of 350 patients, how many patients will elope in a day if the daily probability of elopement is 0.05? If a nursing home takes care of 350 patients, how many patients will elope in a day if the daily probability of elopement is 0.05?

16 Independent Repeated Bernoulli Trials Independence means that the probability of occurrence does not change based on what has happened in the previous day Independence means that the probability of occurrence does not change based on what has happened in the previous day Patient elopes No event Patient elopes No event Patient elopes No event Day 1Day 2Day 3

17 Geometric Probability Density Function Number of trials till first occurrence of a repeating independent Bernoulli event Number of trials till first occurrence of a repeating independent Bernoulli event K-1 non- occurrence of the event occurrence of the event

18 Geometric Probability Density Function Expected number of trials prior to occurrence of the event Expected number of trials prior to occurrence of the event

19 Exercise No medication errors have occurred in the past 90 days. What is the daily probability of medication error in our facility? No medication errors have occurred in the past 90 days. What is the daily probability of medication error in our facility? The time between patient falls was calculated to be 3 days, 60 days and 15 days. What is the daily probability of patient falls? The time between patient falls was calculated to be 3 days, 60 days and 15 days. What is the daily probability of patient falls?

20 Binomial Probability Distribution Independent repeated Bernoulli trials Independent repeated Bernoulli trials Number of k occurrences of the event in n trials Number of k occurrences of the event in n trials

21 Repeated Independent Bernoulli Trials Probability of exactly two elopement in 3 days On day 1 and 2 not 3p p (1-p) On day 1 not 2 and 3p (1-p) p On day 2 3 and not 1P p (1-p)

22 Binomial Probability Distribution Possible ways of getting k occurrences in n trials n! is n factorial and is calculated as 1*2*3*…*n

23 Binomial Probability Distribution k occurrences of the even Possible ways of getting k occurrences in n trials

24 Binomial Probability Distribution n-k non- occurrence of the event k occurrences of the even Possible ways of getting k occurrences in n trials

25 Binomial Density Function for 6 Trials, p=1/2 The expected value of a Binomial distribution is np. The variance is np(1-p)

26 Binomial Density Function for 6 Trials, p=0.05

27 Exercise If the daily probability of elopement is 0.05, how many patients will elope in a year? If the daily probability of elopement is 0.05, how many patients will elope in a year?

28 Exercise 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? What is the probability of 1 or more deaths in 4 months? 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? What is the probability of 1 or more deaths in 4 months? Number of trials =30 Daily probability =0.002 Number of deaths =0 Probability of 0 deaths =0.942 Probability of 1 or more deaths=0.058

29 Exercise 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? What is the probability of 1 or more deaths in 4 months? 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? What is the probability of 1 or more deaths in 4 months? Number of trials =30 Daily probability =0.002 Number of deaths =0 Probability of 0 deaths =0.942 Probability of 1 or more deaths=0.058

30 Exercise Which is more likely, 2 patients failing to comply with medication orders in 15 days or 4 patients failing to comply with medication orders in 30 days. Which is more likely, 2 patients failing to comply with medication orders in 15 days or 4 patients failing to comply with medication orders in 30 days.

31 Poisson Density Function Approximates Binomial distribution Approximates Binomial distribution Large number of trials Large number of trials Small probabilities of occurrence Small probabilities of occurrence

32 Poisson Density Function Λ is the expected number of trials = n p k is the number of occurrences of the sentinel event e = , the base of natural logarithms

33 Exercise What is the probability of observing one or more security violations. when the daily probability of violations is 5% and we are monitoring the organization for 4 months What is the probability of observing one or more security violations. when the daily probability of violations is 5% and we are monitoring the organization for 4 months What is the probability of observing exactly 3 violations in this period? What is the probability of observing exactly 3 violations in this period?

34 Take Home Lesson Repeated independent Bernoulli trials is the foundation of many distributions

35 Exercise What is the daily probability of relapse into poor eating habits when the patient has not followed her diet on January 1 st, May 30 th and June 7 th ? What is the daily probability of relapse into poor eating habits when the patient has not followed her diet on January 1 st, May 30 th and June 7 th ? What is the daily probability of security violations when there has not been a security violation for 6 months? What is the daily probability of security violations when there has not been a security violation for 6 months?

36 Exercise 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? 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? If viruses infect computers at a rate of 1 every 10 days, what is the probability of having 2 computers infected in 10 days? If viruses infect computers at a rate of 1 every 10 days, what is the probability of having 2 computers infected in 10 days?

37 Exercise Assess the probability of a sentinel event by interviewing a peer student. Assess the time to sentinel event by interviewing the same person. Are the two responses consistent? Assess the probability of a sentinel event by interviewing a peer student. Assess the time to sentinel event by interviewing the same person. Are the two responses consistent?