Chapter 17

1. Chapter 17 Probability Models The Binomial Distribution

2. A binomial variable is a variable whose domain contains the number of successes observed in repeated trials of a given experiment. The binomial setting satisfies the following four conditions: Each observation falls into one of just two categories, “success” or “failure.” The probability of success, called p, is the same for each observation. The n observations are all independent. That is, knowing the result of one observation tells you nothing about the other observations. There is a fixed number, n, of observation (trial). The Binomial Distribution

3. The distribution of the X of successes in the binomial distribution with parameters n and p. The parameter n is the number of observations, and p is the probability of a success on any one observation. The possible values of X are the whole numbers from 0 to n. As an abbreviation, we say that X is B(n, p); that is to say that the number of success are defined by our random variable X described by the binomial distribution given n observations with each success having a probability of p. Binomial Distribution

4. An archer has determined that the probability that she hits a bull’s eye is 0.72 on every shot. What is the probability that she will hit exactly 8 bull’s eyes in 10 shots? Let X = random variable whose values are the number of bull’s eyes she could hit on her next 10 shorts. Does this situation satisfy our assumptions? Are there only two outcomes? Yes, either she hits or misses Is the probability the same for each shot? Yes, we are told that the probability of each shot Are her shots independent? Although it is possible that one shot may influence the next (she may adapt), it is acceptable to assume that each shot is independent unless there is a reason not to make this assumption. When in doubt, leave a disclaimer. Example

5. An archer has determined that the probability that she hits a bull’s eye is 0.72 on every shot. What is the probability that she will hit exactly 8 bull’s eyes in 10 shots? Let X = random variable whose values are the number of bull’s eyes she could hit on her next 10 shorts. X is a binomial random variable and we write X: B(10, 0.72) In the calculator, use the function binompdf (n, p) If you use binompdf (10, .72), it will give you the entire probability distribution for this event. For a specific probability (the probability that the archer hits exactly 8 of the 10 shots) use binompdf (n, p, x): binompdf (10, .72, 8) = .2548 Binomial distributions allow us to find the probabilities of these specific types of problems without having to do the tree diagrams. They are intended to save us time and energy. Example

6. Given a discrete random variable X, the probability distribution function assigns a probability to each value of X. The probabilities must satisfy probability rules. Try to find the probability distributions for the following scenarios: A basketball player hits .80 of his free-throws and he shoots six baskets. A child desires a certain toy that is randomly distributed within cereal boxes with a 20% chance of obtaining the particular toy and eight boxes are opened. Try to find the specific probabilities of the following: A basketball player hits .80 of his free-throws and he shoots six baskets. Find the probability that he makes exactly 4 baskets. pdf - probability distribution function

7. cdf-cumulative distribution function • In applications we frequently want to find the probability that a random variable takes a range of values. The cumulative binomial probability is useful in these cases. • cdf - Given a random variable X, the cumulative distribution function (cdf) of X calculates the sum of probabilities for 0, 1, 2, ….., up to the value X. That is, it calculates the probability of obtaining at most X successes in n trials. • It is important to note that the binomcdf (n, p, X) function adds up all the probabilities from 0 to X.

8. The Binomial Coefficient • ABinomial Coefficient counts the number of ways of arranging k successes among n observations and is given by the formula: For k = 0, 1, 2, ……., n. In other words, this gives the number of combinationsfor each event. For example, it can count the number of ways you can pick a group of three people from this class. The number of students would be n and k = 3.

9. The formula for binomial coefficients uses the factorial notation. n! = n x (n-1) x (n - 2) x …….x 3 x 2 x 1. (note: 0! = 1) 4! = 4 x 3 x 2 x 1 = 24 For example, let’s say you want to pick three people from this class and there are 20 students present. Then n = 20 and k = 3. There are 1140 different groups of three that I can choose from a group of 20 people.

10. Binomial Probability If X has the binomial distribution with n observations and probability p of success on each observation, the possible values of X are 0, 1, 2, …, n. If k is any one of these values, P(X = k) = pk (1 - p)n-k Example: Lets say an archer hits the “bulls-eye” 72% of the time that she shoots. Using the Binomial distribution, determine the number ways that she can hit the bulls-eye exactly 4 out of 6 times. P(X = 4) = (.72)4(.28)2 = 0.3160 Or you can you binompdf (n, p, x) = binompdf (6, .72, 4)

11. If a count X has the binomial distribution with number of observations n and probability of success p, the mean and standard deviation of X are µ = np  = These formulas are good only for binomial distributions. Mean and Standard Deviation of a Binomial Random Variable

12. Back to the bulls-eye: The archer hits the “bulls-eye” 72% of the time that she shoots. She takes 6 shots. What is her Expected Value or the mean number of bulls-eyes we expect her to get? What is the standard deviation? Formulas: Mean: µ = np Standard deviation: σ = So, the E(X) = µX = (6)(.72) = 4.32 σX = The mean number of bulls-eyes in six shots is 4.32 with a standard deviation of 1.0998.

13. Binomial Distributions • Suppose an Olympic archer is able to hit the bulls-eye 87% of the time. If she shoots eight arrows, what is the probability that she will get: • (Assume that each shot is independent.) • exactly 4 bulls-eyes • at most 4 bulls-eyes • at least 4 bulls-eyes • What is the Expected Value? • What is the Mean and standard deviation? binompdf (8, .87, 4) ≈ 0.0115 binomcdf (8, .87, 4) ≈ 0.0129 1 – binomcdf (8, .87, 3) ≈ 0.9985 np = 8(0.87) ≈ 6.96 Since E(X) = μ=np = 8(0.87) ≈ 6.96

14. Suppose that a count X has a binomial distribution with n trials and success probability p. When n is large, the distribution of X is approximately normal, The Success/Failure Condition: we will use the normal approximation when n and p satisfy: np 10 and n(1- p)  10. Basically, we can use the normal model if we expect to see at least 10 successes and 10 failures, based on the probabilities, of course. Normal Approximation for Binomial Distributions

15. Example • Use a normal distribution to estimate the following: A baseball player has a .300 batting average. What is the probability that he will hit at least 40 of the next 100 at bats? • First determine if the binomial setting is appropriate. • Now find E(X) • Find σX • Find the z-score for 40 • Determine the P(X ≥ 40) Yes, all of the conditions are met. E(X) = np = 100(.300) = 30 normalcdf(2.18,E99) ≈ 0.01463

16. Assignment