Chapter 8 The Binomial Geometric Distributions 8 1
Chapter 8 The Binomial & Geometric Distributions
8. 1 The Binomial Distribution Definition: “The Binomial Setting” : A situation is said to be a “BINOMIAL SETTING”, if the following four conditions are met: 1. Each observation is one of TWO possibilities - either a success or failure. 2. There is a FIXED number (n) of observations. 3. All observations are INDEPENDENT. 4. The probability of success (p), is the SAME for each observation. n
8. 1 The Binomial Distribution n Definition: “Binomial Distribution” The distribution of the count X of successes in the binomial setting is the BINOMIAL DISTRIBUTION with parameters n and p. n n = the number of observations n p = the probability of success on any one observation n A way to symbolically say this: B(n, p) n
8. 1 The Binomial Distribution n Example 8. 1: BLOOD TYPES n Example 8. 2: DEALING CARDS n Example 8. 3: INSPECTING SWITCHES n Example 8. 4: AIRCRAFT ENGINE RELIABILITY
8. 1 The Binomial Distribution n Finding Binomial Probabilities We will use the TI-83/4 n We will use a “by-hand” formula n n Example 8. 5: INSPECTING SWITCHES SRS of 10 switches from a LARGE shipment n 10% of the switches are “bad” n P(No more than 1 of the 10 switches are “bad”) n Draw a Probability histogram (on TI-8 X) n Binompdf(n, p, X) and Binomcdf(n, p, X) n
8. 1 The Binomial Distribution n Example 8. 6: CORRINE’S FREE THROWS 75% lifetime free-thrower n 12 shots in a key game were takes, and ONLY 7 made … Is this “unusual”? n FIST? n P(X<=7) = ? n
8. 1 The Binomial Distribution n Example 8. 7: THREE GIRLS n n n n Find P(X = 3) L 1 = {0, 1, 2, 3} L 2 = binompdf (3, . 5, L 1) Plot 1…On…Histogram Xlist: L 1 Freq: L 2 WINDOW: Xmin: -. 5 Xmax: 3. 5 Ymin: -. 1 Ymax: . 4 Xlist: L 1 Freq: L 3 = binomcdf (3, . 5, L 1) WINDOW: Ymax: 1. 1 Graph
8. 1 The Binomial Distribution n Example 8. 8: IS CORINNE IN A SLUMP? n n n n n Same 75% free-thrower Let’s create both the probability distribution and the cumulative distribution functions. L 1 = {0, 1, 2, … 10, 11, 12} L 2 = binompdf (12, . 75, L 1) L 3 = binomcdf (12, . 75, L 1) Xlist: L 1 Freq: L 2 WINDOW: Xmin: -. 5 Xmax: 12. 5 Ymin: -. 1 Ymax: . 3 Xlist: L 1 Freq: L 3 = binomcdf (3, . 5, L 1) WINDOW: Ymax: 1. 1 Graph
8. 1 The Binomial Distribution n Example 8. 9: INHERITING BLOOD TYPE Each child in a family has probability of. 25 of having blood type O. n P(X = 2) n FIST? n List by hand all S-F configuration for 2 S’s in a family of 5. n Find each probability … multiply by how many ways it an occur n
8. 1 The Binomial Distribution n n The binomial coefficient: An alternative to listing all 10 options from the previous example. The number of ways of arranging k successes among n observations is given by: Example:
8. 1 The Binomial Distribution n n The Binomial Probability Formula 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, Example 8. 10: DEFECTIVE SWITCHES Part 2
8. 1 The Binomial Distribution n The Binomial Mean and Standard Deviation Example 8. 11: DEFECTIVE SWITCHES Part 3
8. 1 The Binomial Distribution n The Normal Approximation to the Binomial Distribution – when n is “large” … Rule of Thumb: Example 8. 12: ATTITUDES TOWARDS SHOPPING Sample size n = 2500; p =. 6 “Agree – I like buying new clothes, but shopping is often frustrating and time-consuming” P(X >= 1520) 1 – binomcdf(2500, . 6, 1519) … or … Get mean, standard deviation, and then z, and normalcdf
8. 1 The Binomial Distribution n Simulating Binomial Experiments n Example 8. 14: CORINNE’S FREE THROWS n n n p =. 75 … n = 12 … P(X <= 7) = 0. 1576 rand. Bin(1, . 75, 12) L 1: sum(L 1) Simulate 20 games … Compare to. 1576 Get class average. Does Law of Large numbers take over?
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