1. Special Discrete Distributions

2. Bernoulli Trials The basis for the probability models we will examine in this chapter is the Bernoulli trial. We have Bernoulli trials if: –there are two possible outcomes (success and failure). –the probability of success, p, is constant. –the trials are independent.

3. The Binomial Model A Binomial model tells us the probability for a random variable that counts the number of successes in a fixed number of Bernoulli trials. Two parameters define the Binomial model: n, the number of trials; and, p, the probability of success. We denote this Binom(n, p).

4. Binomial Distribution B(n,p) Each trial results in one of two mutually exclusive outcomes. (success/failure)B Outcomes of different trials are independentI NThere are a fixed number of trials N The probability that a trial results in success is the same for all trialsS The binomial random variable x is defined as the number of successes out of the fixed number of Bernoulli trials. Therefore the Binomial Dist. is a Discrete Distribution

5. Are these binomial distributions? 1)Toss a coin 10 times and count the number of heads Yes 2)Deal 10 cards from a shuffled deck and count the number of red cards No, probability does not remain constant 3)Two parents with genes for O and A blood types are starting a family. Count the number of children with blood type O No, no fixed number

6. Toss a coin 3 times and count the number of heads Find the discrete probability distribution X0123X0123 P(x).125.375.375.125 Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads?

7. Binomial Formula: Where the Binomial Coefficient determines the number of ways to choose k events from N

8. Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads?

9. The number of inaccurate gauges in a group of four is a binomial random variable. If the probability of a defect is 0.1, what is the probability that only 1 is defective? More than 1 is defective?

10. Calculator Binomialpdf(n,p,x) – this calculates the probability of a single binomial P(x = k) Binomialcdf(n,p,x) – this calculates the cumulative probabilities from P(0) to P(k) (calculates cumulative probability from left to right only)

11. A genetic trait of one family manifests itself in 25% of the offspring. If eight offspring are randomly selected, find the probability that the trait will appear in exactly three of them. At least 5?

12. In a certain county, 30% of the voters are Republicans. If ten voters are selected at random, find the probability that no more than six of them will be Republicans. Is the independence criteria met in this example?

13. Independence One of the important requirements for Bernoulli trials is that the trials be independent. We have nothing to worry about if the population is infinite. However, when we don’t have an infinite population, the trials are not independent. Luckily, there is a rule that allows us to pretend we have independent trials: –The 10% condition : Bernoulli trials must be independent. If that assumption is violated, it is still okay to proceed as long as the sample is smaller than 10% of the population.

14. BACK TO THE PROBLEM BECAUSE the population of Republicans is so large, drawing a sample of 10 individuals barely changes the probability of each time since 10 is certainly less than 10% of population of Republicans so safe to assume independence P(x < 6) = binomcdf(10,.3,6) =.9894

15. Binomial formulas for mean and standard deviation

16. In a certain county, 30% of the voters are Republicans. How many Republicans would you expect in ten randomly selected voters? What is the standard deviation for this distribution?

17. In L1 – seq(x,x,0,10) In L2 – binompdf(10,.1,L1) Sketch histogram Next go back & increase the probability of success to.3,.5,.7,.9 Compare the resulting histograms

18. What happened to the shape of the distribution as the probability of success increased? What happened to the shape of the distribution as the probability of success increased? As the probability of success increases, the shape changes from being skewed right to symmetrical at p =.5 to skewed left.

19. Calculate the mean and standard deviations for each of the probabilitiesCalculate the mean and standard deviations for each of the probabilities What do you notice? As the probability of success increase, the means increase. the standard deviations increase to p =.5, then decrease. Their values are also symmetrical.

20. The Normal Model to the Rescue! When dealing with a large number of trials in a Binomial situation, making direct calculations of the probabilities becomes tedious (or outright impossible). Fortunately, the Normal model comes to the rescue…

21. The Normal Model to the Rescue (cont.) –Success/failure condition : A Binomial model is approximately Normal if we expect at least 10 successes and 10 failures: np ≥ 10 and nq ≥ 10 As long as the Success/Failure Condition holds, we can use the Normal model to approximate Binomial probabilities.

22. Continuous Random Variables When we use the Normal model to approximate the Binomial model, we are using a continuous random variable to approximate a discrete random variable. So, when we use the Normal model, we no longer calculate the probability that the random variable equals a particular value, but only that it lies between two values and we calculate the mean & standard deviation of the distribution using the Binomial Formulas

23. Geometric Distributions: There are two mutually exclusive outcomes Each trial is independent of the others The probability of success remains constant for each trial. The random variable x is the number of trials UNTIL the FIRST success occurs.

24. The difference between binomial and geometric properties is that there is NOT a fixed number of trials in geometric distributions! Differences between binomial & geometric distributions

25. Binomial random variables start with 0 while geometric random variables start with 1 Other differences: Binomial distributions are finite, while geometric distributions are infinite

26. Geometric Formulas:

27. Count the number of boys in a family of four children. Binomial: X01234 Count children until first son is born Geometric:... X1234...

28. What is the probability that the first son is the 4 th child born? What is the probability that the first son is born is at most the 4 th child?

29. WHAT’S THE PROBABILITY THAT IT TAKES MORE THAN 4 CHILDREN UNTIL A BOY IS BORN?

30. A real estate agent shows a house to prospective buyers. The probability that the house will be sold to the person is 35%. What is the probability that the agent will sell the house to the third person she shows it to? How many prospective buyers does she expect to show the house to before someone buys the house?

31. In L1 – input numbers 1-20 In L2 – geometpdf(.1,L1) Sketch Find the means & standard deviations Increase & then decrease the probability of success What do you see?

32. more stronglyGeometric distributions are skewed right and become more strongly skewed right as the probability of success increases decreaseMean & standard deviation of the distributions decrease as the probability of success increase

33. Poisson Distributions This distribution deals with the probabilities of rare events that occur infrequently in space, time, distance, area, etc. Examples: The number of accidents that occur per month at a given intersection The number of tardies per semester for a given student The number of runs per inning in a baseball game

34. Properties: The occurrence of a success in any interval is independent of that in any other interval The probability that a success will occur in any interval is the same for all intervals of equal size and is proportional to the size of the interval We observe a discrete number of events in a continuous (fixed) interval.

35. Formulas: X = number of rare events per unit of time, space, etc.  = mean value of X (Greek letter lambda)

36. The number of accidents in an office building during a four-week period averages 2. What is the probability there will be one accident in the next four-week period? What is the probability that there will be more than two accidents in the next four-week period?

37. The number of calls to a police department between 8 pm and 8:30 pm on Friday averages 3.5. What is the probability of no calls during this period? What is the probability of no calls between 8 pm and 9 pm on Friday night? What is the mean and standard deviation of the number of calls between 10 pm and midnight on Friday night? P(X = 0) = poissonpdf(3.5,0) =.0302 P(X = 0) = poissonpdf(7,0) =.0009 From 8:00 until 8:30 is a 30 minute period. From 8:00 until 9:00 is a 60 minute period. Since the period is doubled, you must double the mean amount of calls to keep it proportional!  = 14 &  = 3.742 Be sure to adjust !

38. Examine the histograms of the Poisson distributions – = 2 = 4 = 6 What happens to the shape? What happens to the means? What happens to the standard deviations?

39. As increases The distributions become more symmetrical The means increase The standard deviations increase