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Chapter 8 The Binomial & Geometric Distributions

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: Each observation is one of TWO possibilities - either a success or failure .

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Chapter 8 The Binomial & Geometric Distributions

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  1. Chapter 8The Binomial & Geometric Distributions

  2. 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: • Each observation is one of TWO possibilities - either a success or failure. • There is a FIXED number (n) of observations. • All observations are INDEPENDENT. • The probability of success (p), is the SAME for each observation.

  3. 8.1 The Binomial Distribution • 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 = the number of observations • p = the probability of success on any one observation • A way to symbolically say this: B(n, p)

  4. 8.1 The Binomial Distribution • Example 8.1: BLOOD TYPES • Example 8.2: DEALING CARDS • Example 8.3: INSPECTING SWITCHES • Example 8.4: AIRCRAFT ENGINE RELIABILITY

  5. 8.1 The Binomial Distribution • Finding Binomial Probabilities • We will use the TI-83/4 • We will use a “by-hand” formula • Example 8.5: INSPECTING SWITCHES • SRS of 10 switches from a LARGE shipment • 10% of the switches are “bad” • P(No more than 1 of the 10 switches are “bad”) • Draw a Probability histogram (on TI-8X) • Binompdf(n, p, X) and Binomcdf(n, p, X)

  6. 8.1 The Binomial Distribution • Example 8.6: CORRINE’S FREE THROWS • 75% lifetime free-thrower • 12 shots in a key game were takes, and ONLY 7 made … Is this “unusual”? • FIST? • P(X<=7) = ?

  7. 8.1 The Binomial Distribution • Example 8.7: THREE GIRLS • Find P(X = 3) • L1 = {0, 1, 2, 3} L2 = binompdf (3, .5, L1) • Plot1…On…Histogram • Xlist:L1 Freq:L2 • WINDOW: Xmin: -.5 Xmax: 3.5 Ymin: -.1 Ymax: .4 • Xlist:L1 Freq:L3 L3 = binomcdf (3, .5, L1) • WINDOW: Ymax: 1.1 • Graph

  8. 8.1 The Binomial Distribution • Example 8.8: IS CORINNE IN A SLUMP? • Same 75% free-thrower • Let’s create both the probability distribution and the cumulative distribution functions. • L1 = {0, 1, 2, … 10, 11, 12} • L2 = binompdf (12, .75, L1) L3 = binomcdf (12, .75, L1) • Xlist:L1 Freq:L2 • WINDOW: Xmin: -.5 Xmax: 12.5 Ymin: -.1 Ymax: .3 • Xlist:L1 Freq:L3 L3 = binomcdf (3, .5, L1) • WINDOW: Ymax: 1.1 • Graph

  9. 8.1 The Binomial Distribution • Example 8.9: INHERITING BLOOD TYPE • Each child in a family has probability of .25 of having blood type O. • P(X = 2) • FIST? • List by hand all S-F configuration for 2 S’s in a family of 5. • Find each probability … multiply by how many ways it an occur

  10. 8.1 The Binomial Distribution • 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:

  11. 8.1 The Binomial Distribution • 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

  12. 8.1 The Binomial Distribution • The Binomial Mean and Standard Deviation Example 8.11: DEFECTIVE SWITCHES Part 3

  13. 8.1 The Binomial Distribution • 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

  14. 8.1 The Binomial Distribution • Simulating Binomial Experiments • Example 8.14: CORINNE’S FREE THROWS • p = .75 … n = 12 … P(X <= 7) = 0.1576 • randBin(1, .75, 12) • randBin(1, .75, 12)L1:sum(L1) • Simulate 20 games … Compare to .1576 • Get class average. Does Law of Large numbers take over?

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