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Binomial vs. Geometric Distributions

Binomial vs. Geometric Distributions. Chapter 8 Binomial Means and Geometric Distributions. Chapter 8. Binomial Distributions The Binomial setting Probability distribution function (pdf) vs. Cumulative distribution function (cdf) Binomial Coefficient, binomial probability

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Binomial vs. Geometric Distributions

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  1. Binomial vs. Geometric Distributions Chapter 8 Binomial Means and Geometric Distributions

  2. Chapter 8 • Binomial Distributions • The Binomial setting • Probability distribution function (pdf) vs. Cumulative distribution function (cdf) • Binomial Coefficient, binomial probability • Geometric Distributions • Mean, Standard deviations of a binomial / geometric random variable

  3. Binomial vs. Geometric The Binomial Setting The Geometric Setting • Each observation falls into • one of two categories. 1. Each observation falls into one of two categories. • The probability of success • (p) is the same for each • observation. 2. The probability of success (p) is the same for each observation. • The observations are all • independent. • The observations are all • independent. • There is a fixed number (n) • of observations. 4. The variable of interest is the number of trials required to obtain the 1st success.

  4. Binomial Distributions • Some examples… • Are these binomial distributions? • If both sets of parents carry O and A blood type genes, what is the likelihood of getting 2 children with O-genes? • You deal 11 cards to play Gin Rummy, what is the likelihood the 2nd, 3rd or 4th card is a red card? • Computer microprocessor chips have a 99.9% high quality rating, what is the likelihood of getting a fully working chip in a sample of 1000 chips? BINOMIAL: 2 categories ? Constant ‘p’ ? Independent ? Fixed # of events

  5. Binomial Distributions • Some examples… • Assume 2 parents both carry genes for O and A blood type, and that all children from these parents inherit independently of each other. • Consider the children’s blood type from these parents… • Likelihood of getting 2 O-genes (O-type) is 25% (p){possible outcomes: O-O, O-A, A-O, A-A} • If the parents have 4 children (n = 4), the Binomial Distribution B(n,p) can be written as: B(4,0.25)we’ll come back to this more… BINOMIAL: 2 categories ? Constant ‘p’ ? Independent ? Fixed # of events

  6. Binomial Distributions • Some examples… • Deal 11 cards to play Gin Rummy and count the number of red cards… • In this example, there are 11 observations, each with the likelihood of getting a red or black card. • However, once the 1st card is dealt, the likelihood of the 2nd card is not equal amongst red and black. • Since the number of red cards is not necessarily independent (from 1 card to the next), this is NOT a binomial distribution. BINOMIAL: 2 categories ? Constant ‘p’ ? Independent ? Fixed # of events

  7. Binomial Distributions • Some examples… • Manufacturers are constantly evaluating the quality of the products they make. Generally this ‘quality’ is measured by the % likelihood of a product (part) performing to satisfaction. • Assume a computer microprocessor has a probability of 0.999 (99.9%) of performing properly over a week’s worth of time, and that a test sample of 1000 microprocessors is taken and tested for a week. • Since we assume all 1000 of these processors are independent, this can be written as a Binomial Distribution B(1000,0.999) BINOMIAL: 2 categories ? Constant ‘p’ ? Independent ? Fixed # of events

  8. Binomial Distributions • Binomial Distribution definition • The distribution of count X of successes in the binomial setting is the Binomial Distribution with parameters n and p, where • n = # of observations and • p = probability of a success of any 1 observation • Notation: B(n,p) BINOMIAL: 2 categories ? Constant ‘p’ ? Independent ? Fixed # of events

  9. Binomial Probabilities • pdf vs cdf • Probability distribution function (pdf) – Given a discrete random variable X, the pdf assigns a probability to each value of X(i.e., specific probability, X = 2) • Cumulative distribution function (cdf) – Given a discrete random variable X, the cdf calculates the sum of the probabilities for 0,1,2…up to the value X(i.e., at most X successes in n trials, X <= 2)

  10. Binomial Probabilities • Returning to our earlier example… • Likelihood of getting O-type gene is 25% (p){possible outcomes: O-O, O-A, A-O, A-A} • If the parents have 4 children… B(4,0.25) • If we were interested in the likelihood of having 1 out of 4 children with O-type gene… B pdf(4,0.25,1) • => Calculator function… • What is the likelihood of having 2 out of 4 children with O-type gene… B pdf(4,0.25,2) • What is the likelihood of having at most 2 out of 4 children with O-type gene… B cdf(4,0.25,0,2)

  11. Binomial Probabilities • Looking at the Binomial Activity worksheet… • Likelihood of a couple having 3 girls consecutively • Binomial setting? • Notation B(3,0.50) • B pdf(3,0.50,3) • Likelihood of getting O-type gene is 25% (p){possible outcomes: O-O, O-A, A-O, A-A} • If the parents have 4 children… B(4,0.25)

  12. Binomial Distributions • Text Example from p.443… • Let’s assume Corinne typically makes 75% of her free throws in a game. In a recent game, she makes only 7 of 12 FTs and fans wonder if this is a ‘reasonable’ performance… • First, is this a binomial setting? • If so, is this a “pdf” or “cdf” calculation?(hint: she makes at most 7 FTs…)

  13. Binomial Distributions • Example from p.443… • Corinne: typically 75% FTs, makes 7 of 12… • Assume all FTs are independent, X=made FT, the Binomial Distribution can be reflected as B(12,0.75) • P(X 7) = P(X=0) + P(X=1) + … P(X=7)B cdf (12,0.75,7) => 0.1576 (15.76%) • SO, according to the binomial distribution, Corinne will make at most 7 of 12 FTs about 16% of the time. Although it is a low percentage, it is quite possible she would make only 7 of 12 free throws.

  14. Binomial Probability • So far, we’ve worked with small numbers to calculate the binomial distributions and probabilities. For larger values and combinations, calculating probabilities can be cumbersome… • More generically, the binomial probability can be calculated as:

  15. Binomial Probability • The Binomial Probability formulahas 2 components… • Binomial coefficient… which is the total number of combinations of a given situation • Probability calculation… which incorporates the probability values B pdf – Menu 5-5, D… B cdf – Menu 5-5, E…

  16. Binomial Coefficient • The number of ways of arranging ‘k’ occurrences among ‘n’ observations is given by the Binomial Coefficient: • For example, 1 occurrence of 4 observations (i.e., 1 in 4) would be written as: Formula:

  17. Binomial Coefficient • Some examples… Formula:

  18. Binomial Coefficient • A ‘shortcut’…  • For the numbers 6 and 4, notice that 6*5 remains on top (i.e., all numbers > 4) and 2! remains on the bottom (i.e., 6 – 4) Formula:

  19. Binomial Coefficient • More for fun… Formula: Menu 5-3…

  20. Binomial Probability • Binomial Probability • Returning to our couple desiring 3 children… • Using the formula above, calculate the probability of having 2 girls among 3 children NOTE: Don’t forget the combinations (nCr) term! (e.g., 3 in this case)

  21. Binomial Probability • 1 more example for the couple… • Calculate the probability of having at least 2 girls • Note: In this particular case, you could also use B cdf (3,0.50,2,3)…

  22. Developing the Formula • Assume an event O occurs 25% of the time(therefore the event Oc occurs 75%) • The probability distribution would be… This is in the format of…Combinations *(p)k(1-p)n-k

  23. Developing the Formula n = # of observations p = probablity of success k = given value of variable Rewritten Use the Binomial Coefficientto determine the # of combinations

  24. Are Random Variables and Binomial Distributions Linked? • Remembering our example of customers buying electric vs. gas hot tubs from before (typically, 40% buy electric hot tubs), the probability distribution looked like… X = number of people who purchase electric hot tubs X 0 1 2 3 (.6)(.6)(.6) GGG EEG GEE EGE (.4)(.4)(.6) (.6)(.4)(.4) (.4)(.6)(.4) .288 P(X) .216 .432 .064 EGG GEG GGE (.4)(.6)(.6) (.6)(.4)(.6) (.6)(.6)(.4) EEE (.4)(.4)(.4)

  25. Binomial Distributions • Continuing with our example of customers buying electric vs. gas hot tubs… X = number of people who purchase electric hot tubs X 0 1 2 3 (.6)(.6)(.6) GGG EEG GEE EGE (.4)(.4)(.6) (.6)(.4)(.4) (.4)(.6)(.4) .288 P(X) .216 .432 .064 EGG GEG GGE (.4)(.6)(.6) (.6)(.4)(.6) (.6)(.6)(.4) EEE (.4)(.4)(.4)

  26. Confirming the Formula n = # of observations p = probablity of success k = given value of variable How many combinations of customers picking 2 gas hot tubs? (.6)(.6)(.6) GGG EEG GEE EGE (.4)(.4)(.6) (.6)(.4)(.4) (.4)(.6)(.4) EGG GEG GGE (.4)(.6)(.6) (.6)(.4)(.6) (.6)(.6)(.4) EEE (.4)(.4)(.4)

  27. Working with Probability Distributions • State the distribution to be used • Define the variable • State important numbers • Binomial: n & p • Geometric: p Key steps for all probability distributions

  28. Some Examples… • Twenty-five percent of the customers entering a grocery store between 5-7 PM use an Express Checkout. • Consider five randomly selected customers, and let X denote the number among the five who use the express checkout. • Binomial, n = 5, p = 0.25 B(5,0.25)X = # of people use Express Checkout

  29. Some Examples… • What is the probability that two used express checkout? • Binomial, n=5, p=0.25, k = 2 B(5,0.25,2)X = # of people use Express Checkout 26.37% likelihood of finding 2 customers in 5 using Express Checkout. 10 combinations

  30. Some Examples… • What is the probability that at least four used express checkout? Binomial n = 5, k = 4 or 5, p = 0.25 B(5,0.25)X = # of people use Express Checkout 5 combinations 1 combination

  31. Some Examples… • The following question was asked to a national sample of American adults with children in a Time/CNN poll (Jan96)“Do you believe your children will have a higher standard of living than you?” • Assume that the true percentage of all American adults who believe their children will have a higher standard of living is 0.60. • Let X represent the number who believe their children will have a higher standard of living from a random sample of 8 American adults. Binomial, n = 8, p = 0.60 B(8,0.60)X = # of people who believe…

  32. Some Examples… • “Do you believe your children will have a higher standard of living than you?”Interpret P(X=3) and find the numerical answer… • ANSWER: The probability that (exactly) 3 people from the random sample of 8 believe their children will have a higher standard of living. Binomial, n=8, p=0.60, k=3 B(8,0.60,3)X = # of people who believe… 56 combinations

  33. Some Examples… • “Do you believe your children will have a higher standard of living than you?” • Find the probability that none of the parents believe their children will have a higher standard. Binomial, n=8, p=0.60, k=0 B(8,0.60,0)X = # of people who believe… 1 combination

  34. Chapter 8 • Binomial Distributions • Geometric Distributions • The Geometric setting • Calculating geometric probability • Mean, Standard deviations for binomial / geometric random variable

  35. Binomial vs. Geometric The Binomial Setting The Geometric Setting • Each observation falls into • one of two categories. 1. Each observation falls into one of two categories. • The probability of success • is the same for each • observation. 2. The probability of success is the same for each observation. • The observations are all • independent. • The observations are all • independent. • There is a fixed number n • of observations. 4. The variable of interest is the number of trials required to obtain the 1st success.

  36. Geometric Distributions • The Geometric Setting • “The variable of interest is the number of trials required to obtain the 1st success” • Note: It doesn’t have to be the 1st occurrence of something, just the 1st “success”… and we (or the question) can determine success. Some examples: • Flip a coin until you get a head • Roll a dice until you get a particular number • Attempt a 3-point shot until 1 is made • Turn over cards until a face card is displayed

  37. Geometric Distributions • The Geometric Setting • Class example: Rolling a dice until a 6 appears… • What is the likelihood a 6 appears on the 1st roll? • On the 2nd roll? • On the 4th roll?

  38. Developing the Geometric Formula Probability X Result on the 1st occurrence Result on the 2nd occurrence Result on the 3rd occurrence Result on the 4th occurrence

  39. Geometric Distributions • Geometric Distribution definition • If X has a Geometric Distribution with probability p of success and (1-p) of failure, the probability that the 1st success occurs on the nth trial is • n = # of trials required to achieve success • p = probability of a success of any 1 observation • Notation: G(n,p) • With Geometric Distribution formula G pdf – Menu 5-5, F… G cdf – Menu 5-5, G…

  40. Geometric Distributions • Examples… • Suppose we have data that suggest that 3% of a company’s hard disc drives are defective. You have been asked to determine the probability that the first defective hard drive is the fifth unit tested (i.e., not till the 5th hard drive). • Distribution: G(n,p) => G(5,0.03)

  41. Geometric Distributions • Examples… • A basketball player makes 80% of her free throws. We put her on the free throw line and ask her to shoot free throws until she misses one. • Distribution: G(n,p) => G(?,0.20)X = number of free throws the player takes until she misses

  42. Geometric Distributions • Examples… • X = number of free throws the player takes until she misses • What is the probability that she will make 5 shots before she misses? • Geometric, missed 6th shot, G pdf (0.20,6)

  43. Geometric Distributions • Examples… • Y = number of free throws the player takes until she makes one • What is the probability that she will miss 5 shots before she makes one? • Geometric, made on 6th shot, G pdf (0.80,6)

  44. Geometric Distributions • Examples… • Y = number of free throws the player takes until she makes one • What is the probability that she will make at most 5 shots before she misses? • Geometric, made on 6th shot, G cdf (0.20,1,6) 0.20 + 0.80*0.20 + 0.802*0.20 + 0.803*0.20 + 0.804*0.20 + 0.805*0.20 =0.7379

  45. Distributions • Are the following distributions Binomial, Geometric, or neither? If Binomial/Geometric, what are the key variables and what is the distribution formula? • Let’s say former Houston Astros Hunter Pence typically hits on average 0.280. What is the probability in the 1st game of the season that he hits 2 out of 4 at bats? • Binomial B(4,0.280), B pdf (4,0.280,2)

  46. Distributions • Are the following distributions Binomial, Geometric, or neither? If Binomial/Geometric, what are the key variables and what is the distribution formula? • Continuing with Hunter Pence’s 0.280 batting average…In that same game, what is the probability he gets his 1st hit on the 2nd at bat? • Geometric G(2,0.280), G pdf (0.280,2)where X = # of hits

  47. Distributions • Are the following distributions Binomial, Geometric, or neither? If Binomial/Geometric, what are the key variables and what is the distribution formula? • Wal-Mart customers typically return products immediately after the holidays about 20% of the time. What is the likelihood the 3rd customer coming into the store has a return item? • Geometric G(3,0.20), G pdf (0.20,3)where X = customers returning items

  48. Distributions • Are the following distributions Binomial, Geometric, or neither? If Binomial/Geometric, what are the key variables and what is the distribution formula? • Wal-Mart customers typically return products immediately after the holidays about 20% of the time. What is the likelihood at least 2 of 10 randomly sampled customers return their products? • Binomial B(10,0.20), B cdf (10,0.20,2)where X = customers returning items

  49. Distributions • Are the following distributions Binomial, Geometric, or neither? If Binomial/Geometric, what are the key variables and what is the distribution formula? • Motorola claims to have a 99.9% high product quality rating on its cell phone microprocessor chips. Sampling 1000 chips from the Austin, TX manufacturing facility, what is the likelihood we find 1 chip which is defective? • Binomial B(1000,0.001), B pdf (1000,0.001,1)where X = # of defective chips

  50. Chapter 8 • Binomial Distributions • Geometric Distributions • The Geometric setting • Calculating geometric probability • Mean, Standard deviations for binomial / geometric random variable

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