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Probability

Probability. Definition: randomness, chance, likelihood, proportion, percentage, odds. Not sure what will happen in a single event but, in the long run, certain patterns emerge. Probability is the mathematical ideal. We use letters like X and Y to represent quantities.

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Probability

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  1. Probability Definition: randomness, chance, likelihood, proportion, percentage, odds. Not sure what will happen in a single event but, in the long run, certain patterns emerge. Probability is the mathematical ideal. We use letters like X and Y to represent quantities. These will be called random variables.

  2. Probability Model • List the outcomes for a given event (experiment or question) and associated probabilities. • S: sample space (contains all possible outcomes) • Event: single outcome or collection of outcomes • Example: pick a card out of a standard deck • S: sample space contains • Event: pick a • pick a

  3. Basic Rules • Event A has probability P(A), which is between 0 and 1 (inclusive). • Probability of entire sample space, P(S), is . • Addition: If two events are disjoint (nothing in common), then P(A or B) = . • Complement: P(not A) =

  4. Probability Model for standard deck of cards • 52 cards, 4 suits (Diamonds, Hearts, Clubs, Spades) • Each suit as 13 cards: 2, 3, 4, 5, … , 9, 10, J, Q, K, A • P(picking any single card)= • A = event that a red 5 is picked • B = event that a club is picked • C = event that a face card (J, Q, K) is picked • P(A or B)= • P(not C) =

  5. Discrete Model • If sample space is finite, the probability model is called discrete. • List all outcomes and associated probabilities in a table. • Roll 2 six-sided dice and record the sum.

  6. Continuous Model • If sample space contains a range of values, the probability model is called continuous. • Density curves record probabilityas the area under the • curve for a given range of outcomes. • So, total area under the curve will always equal 1.

  7. Continuous Model – Example 1 • The uniform distribution for any real number, X, from 3 to 7 looks like: 3 5 7 3 5 6 7 3 5 7 5.1 3.2

  8. Continuous Model – Example 2 • The symmetric triangular distribution for any real number, X, from 0 to 8 looks like: 4 8 4 8 4 8

  9. More Probability Use Venn diagrams to visualize probability rules. Sample space, S: rectangle Events (A, B, C, …): circles inside If events are disjoint, don’t overlap circles. Keep track of # of outcomes in each region of the rectangle.

  10. Venn diagram - example • Example: pick a card out of a standard deck • S: sample space contains 52 outcomes (52 cards) • A = event that a red 5 is picked • B = event that a club is picked • C = event that a face card (J, Q, K) is picked • S • B • A • C

  11. P(A or B) • P(B or C) • S • S • B • B • 10 • 10 • A • A • 2 • 3 • 3 • 2 • 9 • 9 • 28 • 28 • C • C

  12. General Addition Rule • A = event that a red card is picked • B = event that a number card is picked • P(A or B) • S • A • B • General Addition: P(A or B) =

  13. Conditional Probability Rule • Given a condition (you know something happened), how • does that change the chances of something else happening? • P(B|A)= probability of B given A • S • A • B

  14. Venn Diagram of 70 students • C: owns a cat • D: owns a dog • 10 • S • D • C • 30 • 20 • 10

  15. General Multiplication Rule • Rewrite • to get: • Experiment: pick two cards out of a standard deck

  16. Independent Events • Two (or more) events are independent if knowledge of one event does not change the chances of the other. • Multiplication Rule for Independent Events:

  17. For a cholesterol-lowering drug, there is a 5% chance that a loss-of-sleep side effect will occur. • What are the chances that two people picked at random take the drug and experience sleep loss? • What are the chances that at least 1out of 3 loses sleep?

  18. The Normal Distribution Use curves to describe overall pattern seen in a histogram. Curve will capture 100% of all observations. Hence, there will be a total area of 1 below it. Then the area under the curve for a given range of values will represent the proportion (percent, fraction) of observations that fall in that range.

  19. Curves and proportions % • The proportion of scores above 80 is roughly 26.8%. v 40 60 20 80 100 % • The area under the density curve for scores above 80 is roughly 0.261 =26.1%. v 40 60 20 80 100

  20. Mean and Medians • Location of the median on a density curve is where area under is cut in half. • Location of the mean on a density curve is where the length of the curve is cut in half. • On symmetric curves: • On skewed curves:

  21. Normal curves are special kinds of density curves • Symmetric, single peaked, bell-shaped • Use m(mu) ands(sigma)to talk about mean and std. dev. • – • s – m m - s m + s

  22. 68-95-99.7 Rule • About 68% of data fall within • About 95% of data fall within • About 99.7% of data fall within m m + 2s m + 3s m - 3s m - 2s m - s m + s

  23. Example 1 • Grasshopper jumps can be described by a Normal • distribution with m = 12 inches and s = 2 inches. • About 68% of all jumps are between inches. • About where would you find the top 2.5%? 68% 95% 99.7% 12 16 18 6 8 10 14

  24. Example 1 – continued • What % falls below 14 inches? 12 12 16 18 16 18 8 6 6 8 10 14 10 14 16 18 6 8 10 14 12 • What % of jumps are more than 14 inches? • What % of jumps are between 14 and 16 inches?

  25. Finding values without 68-95-99.7 • We use tables or calculators to find harder values, like where is the top 10% or what percent falls below a given observation. • N(m, s) means observations come from a Normal distribution with a mean of m and a standard deviation of s. • Standardize observationx from N(m, s) by: • The standardized value is called a

  26. z-scores from example 1, N(12, 2)

  27. Two functions on the calculator • (found under 2nd VARS => DISTR) • normalcdf( : will give area between two bounds for a given m, s. • invNorm( : will give the observation that has a particular area to its left for a given m, s. • normalcdf(lower bound, upper bound, m, s) • invNorm(area, m, s) m m - s m + s

  28. Using the calculator with grasshopper N(12, 2) • What % of jumps fall below 17 inches? • No lower bound, so: 12 16 18 8 6 10 14 • normalcdf(lower bound, upper bound, m, s) • = normalcdf( ) = area below 17 = • What % of jumps fall above 11.5 inches? • First, find area 12 16 18 8 6 10 14 • normalcdf(lower bound, upper bound, m, s) • = normalcdf( ) = area = • Since total area is 1 and we have : • we want

  29. Using the table with grasshopper N(12, 2) • What % of jumps fall between 10 and 16.36 inches? • - • = 12 12 16 18 16 18 8 8 6 6 10 14 10 14 12 16 18 8 6 10 14 • Area between = area below 16.36 – area below 10. • Calculator does this all at once with the normalcdf( function. • normalcdf(lower bound, upper bound, m, s) • = normalcdf( ) = area between =

  30. Using the table with grasshopper N(12, 2) • What jumps fell in the top 10%? • 10% 12 16 18 8 6 10 14 • What observation has an area of .10 above it? • What observation has an area of .90 below it? • Use invNorm function to find that observation. • invNorm(area, m, s) • = invNorm( ) = value with .9 area below=

  31. Using the table with grasshopper N(12, 2) • 50% • Where do the middle 50% fall? 12 16 18 8 6 10 14 • What observation has an area of below it? • What observation has an area of below it? • Use invNorm function to find those observations. • invNorm(area, m, s) • = invNorm( ) = value with area below = • invNorm(area, m, s) • = invNorm( ) = value with area below =

  32. 68-95-99.7 Rule • 1,2,3 standard deviations away accurate to two decimal places

  33. Sampling Distributions Know the entire population: (parameter) Know only a sample (SRS): (statistic)

  34. Law of Large Numbers • - As you increase the sample size, sample mean gets closer to population mean • Population = 3, 3, 8, 15, 20, 21, 22, 31, 39 • Sample of size 1= 8 • Sample of size 2= 8, 22 • Sample of size 3= 8, 22, 31 • Sample of size 4= 8, 22, 31, 3 • Sample of size 5= 8, 22, 31, 3, 20

  35. Population of 7 people and their weights (in pounds) • 122, 140, 150, 155, 160, 170, 195 • Samples of size 1: {122}, {140}, {150}, {155}, {160}, {170}, {195} • Mark off the sample mean for each sample with an “x” • x • x • x • x • x • x • x • 120 • 130 • 140 • 150 • 160 • 170 • 180 • 190 • 200 • Samples of size 2: {122, 140}, {122, 150}, {122, 155}, {122, 160}, {122, 170}, {122, 195}, {140, 150}, …, (170, 195}. There are 21 possible samples. • Mark off the sample mean for each sample with an “x” • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • 120 • 130 • 140 • 150 • 160 • 170 • 180 • 190 • 200

  36. Population of 7 people (continued) • 140, 122, 160, 195, 150, 155, 170 • Samples of size 1: • x • x • x • x • x • x • x • 120 • 130 • 140 • 150 • 160 • 170 • 180 • 190 • 200 • Samples of size 2: • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • x • 130 • 140 • 150 • 160 • 170 • 180 • 190 • Samples of size 6: 7 possible sample of this size. {122, 140, 160, 150, 155, 170}, … • x • x • x • x • x • x • x • 140 • 150 • 160 • 170 • 180

  37. Sampling distribution of • Sampling from a large population with mean m and • standard deviation s: • samples of size n will have their sample means distributed • with a mean m and standard deviation s over root n. • If population is N(m, s), then • If population is not Normal but n is large, then

  38. Ex. 1 - Weight of eggs is N(65, 3) • Your egg carton holds 9 eggs, so consider each carton as a random sample of 9 eggs. Let X be the weight of a single egg in grams and X be average weight of your carton. • What is the sampling distribution for your carton’s average weight?

  39. Weight of eggs is N(65, 3) – continuedMean weight of carton is N(65,1) • Convert 67 to a z-score • for the carton: • Convert 67 to a z-score • for a single egg: 67 56 59 68 71 62 65 74

  40. Ex. 2 - Length of trout is N(17.5, 2.5) • Your local waters contain a multitude of trout. Let X be the length of a single fish in inchesandX be average length of your daily catch of five fish. • What is the sampling distribution for your daily catch?

  41. Trout length is N(17.5, 2.5) – continuedMean length of daily catch is N(17.5,1.118) • Convert 16 to a z-score • for the daily catch: • Convert 16 to a z-score • for a single fish: 10 12.5 20 22.5 15 17.5 25

  42. Trout length is N(17.5, 2.5) – continuedMean length of daily catch is N(17.5,1.118) 10 12.5 20 22.5 15 17.5 25 15 10 12.5 17.5 20 22.5 25

  43. Ex 3 - Length of trout is N(10, 2) • Your fishing pond has another type of trout. Let X be the length of a single fish in inches taken at random andX be average length of a sample of 16 fish. • What is the sampling distribution for a sample of 16 fish?

  44. Trout length is N(10, 2) – continuedMean length of 16 fish is N(10,0.5) 8 4 6 12 14 10 16 4 6 8 10 12 14 16

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