2. Second Midterm Exam Friday, April 17 50 minutes You may bring a cheat sheet with you. Charts Regression Solutions to homework 4 are available online.

3. Some concepts Event An observation. e.g., Flip a coin. Outcome Result of an event. e.g., head/tail Probability “Chance” of the occurrence of an event. e.g., ½ to get a head/tail Sample space Set of all possible outcomes of an experiment. e.g., {head, tail}

4. Some more concepts Independent Events The outcome of one event does not influence the outcome of the other. e.g., Flip two coins Mutually Exclusive Events/Outcomes e.g., One coin flip cannot result in both a head and a tail.

5. Probability Priori Probability Known before the event occurs p(head in a coin flip) = ½ p(rolling 1 on a die) = ? Games are usually based on priori probabilities. Empirical Probability Based on actual observations or event occurrences e.g., What is the proportion of blue M&Ms in a bag? e.g., proportion of red/white blood cells? Frequency counts are key to calculating empirical probabilities

6. Examples Download SimpleProbability.xlsx Q1: What is the probability that any family selected at random from among the 2,556 families will have no children? Q2: What is the probability that any family selected at random from among the 2,556 families will have at least 4 children?

7. Some facts The probabilities of mutually exclusive events can be added together. The sum of the probabilities of all possible mutually exclusive probabilities will always be ?.

8. Exercise Lay out on a spreadsheet the possible outcomes of two rolls of a fair die and calculate the following: The probability that two faces of the dice will equal 7 The probability that two faces of the dice will equal 8 or more

9. Exercise Lay out on a spreadsheet the possible outcomes of a visit by three persons to an emergency room. Calculate the following: The probability of no emergencies out of three arrivals The probability of one emergency out of three arrivals The probability of two emergencies out of three arrivals The probability of three emergencies out of three arrivals What is the probability of each outcome?

10. However.. The probability of a visit being an emergency is empirical probability. Download Emergencies.xlsx for historical records.

11. Sequential Events Under the assumption of independence p(E1E2E3...En) = p(E1) * p(E2) * p(E3) *... * p(En)

12. Binomial Probability Binomial is a probability distribution that describes the behavior of a binary event (yes/no, head/tail, emergency/non-emergency, etc) BINOMDIST() function

13. The BINOMDIST() Function = BINOMDIST(k, n, p, cumulative?) where k is the number of emergency visits (3, 2, 1, 0) n is the total number of visits observed (3) p is the probability of an emergency (0.646) and 0 or 1 to indicate whether it is actual probability or the cumulative probability

14. Cumulative Distribution Accumulation of the probabilities What is the probability of having at most one emergency visit? How about at most 2 emergencies? 3 emergencies? Cumulative binomial function always accumulates from the lowest number to the highest.

15. Exercise A local health department counsels patients coming to a clinic on cigarette smoking only if they are smokers. History has shown that about 27% of patients are smokes when they first come to the clinic. Assume that the clinic will see 15 patients today. Graph both the Binomial distribution and the cumulative distribution What is the probability that 10 people are smokers? What is the probability that 10 patients or more are smokers? What is the probability that 5 or fewer patients are smokers? What is the probability that between 7 and 10 patients (inclusive) are smokers?

16. Poisson Distribution The same clinic example: On average there is 0.9 persons every 15 minutes. How often does the nurse have to be prepared to deal with 2 people in any 15 minute interval? How about 3 or 4 people?

17. The POISSON() Function = POISSON(k, λ, cumulative?) where k is the number of arrivals (1, 2, 3,...) λ is the average number of arrivals (0.9) and 0 or 1 to indicate whether it is actual probability or the cumulative probability

18. Example On average there is 0.9 persons every 15 minutes. Graph the Poisson distribution and the cumulative distribution. Experiment to see how many scores you need to include. (Keep 4 decimal places in the values of the probabilities.) What is the probability that in any 15 minute interval, two patients show up? How about 5 or fewer than 5? At least 2?

19. Exercise The hospital supply room manager found that on average about 2 gloves in a box are not usable. Graph the Poisson distribution and the cumulative distribution. Experiment to see how many scores you need to include. (Keep 4 decimal places in the values of the probabilities.) What is the probability that in any given box of gloves, only one will not be usable? How about fewer than 5 will not be usable? At least 2?