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**AP Statistics – Probability**

Mrs. Lerner Charlotte Catholic High School With some updates by Dr. Davidson Mallard Creek High School

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**Random Phenomena Are uncertain in the short run**

Exhibit a consistent pattern in the long run Note the dual aspect Note: This is the Condition for the Probability we study

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Definitions: An event is an outcome or a set of outcomes of random phenomenon (RP). Outcome(s) of interest are “Success(es)” [even if they are bad news!!!!]

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**Probability of an Event :**

(note dual aspect) Probability of an event = proportion of times a success occurs in the long run (measure by experiment or simulation) [number of ways for a success to occur]/[all possible events (specified by a model of the RP)

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**All Possible Outcomes If there are n ways to do a first event**

& m ways to do a second event Then the number of all possible outcomes=nm This is called the multiplication rule

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**Probability – Example # 1**

Consider the sum of two dice that are rolled Let’s say we are interested in the Probability the sum is a 4, so SUCCESS = ? Find # of ways to get a sum = 4 [a success] is 3: , 2+2, [3 ways]

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**Probability The probability of a specific outcome is**

[# of ways to get a success]/ [all possible outcomes e.g. Probability of getting a sum =4 is: 3/36 = 1/12 ~ = 8.33 % This is the chance the next outcome is a success This is the proportion of times that successes occur in a large repetitive # of identical trials

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**The probability P(A) of any event A satisfies 0≤P(A)≤1**

Probability Rules - 1 The probability P(A) of any event A satisfies ≤P(A)≤1 Thus the probability of any event is between 0 & 1

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**The probabilities of all possible outcomes must add up to 1**

Probability Rules - 2 The Sample Space is the set of all possible outcomes If S is the sample space in a probability model, then P(S)=1 The probabilities of all possible outcomes must add up to 1

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**Probability Rules - 3 the event that A does not occur written as Ac**

The complement of any event A is the event that A does not occur written as Ac The complement rule states that P(Ac)=1-P(A) The probability that an event does not occur is 1 minus the probability that it does occur

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**Probability Rules - 4 if they have no outcomes in common**

Two events A and B are disjoint if they have no outcomes in common cannot occur at the same time. If A and B are disjoint, then P(A or B) = P(A) + P(B). If two events have no outcomes in common, then the probability of either one occurring is the sum of their individual probabilities

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**Probability Rules - 5 P(A or B) = P(A) + P(B) – P(A and B)**

For any two events A and B: P(A or B) = P(A) + P(B) – P(A and B) The probability of either A or B occurring is the sum of their individual probabilities minus the probability that they occur at the same time

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Probability Rules - 6 Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, then P(A and B) = P(A)P(B). If two events are independent, then the probability that they both occur is the product of their individual probabilities Note – Disjoint events are always NOT INDEPENDENT, for if one occurs, then we know the other can not occur!

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**Remember… And = Joint = Intersection**

Or = Both (1, or other, or 2) = Union

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**P(B|A)=P(A and B)/P(A)**

Probability Rules - 7 When P(A)>0, the conditional probability of B given A is: P(B|A)=P(A and B)/P(A) The probability of B occurring, given that A occurs, is the probability of A & B jointly occurring divided by the probability of the stated condition (A) occurring

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**P(A and B) = P(B|A) P(A) P(A and B) = P(A|B) P(B)**

Probability Rules - 8 For any two events A and B, P(A and B) = P(B|A) P(A) P(A and B) = P(A|B) P(B) The probability of A and B occurring jointly is the probability that B occurs given that A has already occurred multiplied by the probability that A occurs

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Example - 2: Suppose that for a certain Caribbean island in any 3-year period: the probability of a major hurricane is .25 the probability of water damage is .44 and the probability of both a hurricane and water damage is .22. What is the probability of water damage given that there is a hurricane?

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**Suppose that for a certain Caribbean. island in any 3-year period the**

Suppose that for a certain Caribbean island in any 3-year period the probability of a major hurricane is .25, the probability of water damage is and the probability of both a hurricane and water damage is What is the probability of water damage given that there is a hurricane?

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**If three people, Joe, Betsy, and Sue, play a game in which**

Joe has a 25% chance of winning Betsy has a 40% chance of winning What is the probability that Sue will win?

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**If three people, Joe, Betsy, and Sue, play**

If three people, Joe, Betsy, and Sue, play a game in which Joe has a 25% chance of winning and Betsy has a 40% chance of winning, what is the probability that Sue will win?

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**A summer resort rents rowboats to customers**

A summer resort rents rowboats to customers but does not allow more than four people to a boat. Each boat is designed to hold no more than 800 pounds. Suppose the distribution of the weight of adult males who rent boats, including their clothes and gear, is normal with a mean of 190 pounds and standard deviation of 10 pounds. If the weights of individual passengers are independent, what is the probability that a group of four adult male passengers will exceed the acceptable weight limit of pounds?

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**5. What proportion of the Republicans are over 50?**

The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city. Dem. Repub. Indep. Over _____________________________________________________________ 5. What proportion of the Republicans are over 50? 61/238 32/96 96/238 32/61 Cannot be determined.

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**5. What proportion of the Republicans are over 50?**

The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city. Dem. Repub. Indep. Over _____________________________________________________________ 5. What proportion of the Republicans are over 50? 32/96

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The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city. Dem. Repub. Indep. Over _____________________________________________________________ 6. If one adult citizen is chosen at random, what is the probability that this person is a Democrat between the ages of 41 and 50? 17/238 17/88 61/238 17/61 88/238

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The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city. Dem. Repub. Indep. Over _____________________________________________________________ 6. If one adult citizen is chosen at random, what is the probability that this person is a Democrat between the ages of 41 and 50? 17/238

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The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city. Dem. Repub. Indep. Over _____________________________________________________________ 7. Given that a person chosen at random is between 31 and 40, what is the probability that this person is an Independent? 10/238 10/63 10/54 54/238 63/238

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The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city. Dem. Repub. Indep. Over _____________________________________________________________ 7. Given that a person chosen at random is between 31 and 40, what is the probability that this person is an Independent? 10/63

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**8. What proportion of the citizens sampled are over 50 or Independent?**

The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city. Dem. Repub. Indep. Over _____________________________________________________________ 8. What proportion of the citizens sampled are over 50 or Independent? 54/238 61/238 100/238 115/238 Cannot be determined

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**8. What proportion of the citizens sampled are over 50 or Independent?**

The following table shows the frequencies of political affiliations in the age ranges listed from a random sample of adult citizens in a particular city. Dem. Repub. Indep. Over _____________________________________________________________ 8. What proportion of the citizens sampled are over 50 or Independent? 100/238

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Example - Assume an multiple-choice examination consists of questions, each having five possible answers. Linda estimates that she has probability of knowing the answer to any question that may be asked. If she does not know the answer, she will guess, with conditional probability 1/5 of being correct. What is the probability that Linda gives the correct answer to a question? (Draw a tree diagram to guide the calculations.)

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An examination consists of multiple-choice questions, each having five possible answers. Linda estimates that she has probability of knowing the answer to any question that may be asked. If she does not know the answer, she will guess, with conditional probability 1/5 of being correct. What is the probability that Linda gives the correct answer to a question? (Draw a tree diagram to guide the calculations.) P(correct) = *.2 = .8

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Random Variables A random variable assumes any of several different values as a result of some random phenomenon

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**Discrete Random Variables [RV]**

A Discrete RV – has a countable number of possible values The probabilities must satisfy two requirements Every probability is between 0 and 1 The sum of all probabilities is 1 We can use a probability histogram to look at the probability distribution. Value of X x1 x2 x3 … xk Probability p1 p2 p3 pk

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**Discrete Random Variables**

Mean of a Discrete R. V. – (also called expected value) –

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**Discrete Random Variables**

Variance of a Discrete R. V. –

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**Continuous Random Variables**

Continuous R. V. – takes all values in an interval of numbers We look at its distribution using a density curve The probability of any event is the area under the density curve in that interval.

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**If X and Y are R. V.‘s, then μX±Y = μX ± μY**

Rules for Means If X is an R. V. and a & b are fixed numbers, then the mean μa+bX = a +bμX If X and Y are R. V.‘s, then μX±Y = μX ± μY

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**Rules for Variances square ‘em, add ‘em, take the square root**

If X is an R. V. and a and b are fixed numbers, then σ2a+bX = b2σX2 Note that multiplying by a constant changes the variance but adding a constant does not. If X and Y are independent R. V.’s, then σ2X±Y =σX2+σY “Pythagorean Theorem of Statistics” For STANDARD DEVIATION: square ‘em, add ‘em, take the square root

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11. Suppose X and Y are random variables with μX = 10, σX = 3, μY = 15, and σY = 4. Given that X and Y are independent, what are the mean and standard deviation of the random variable X+Y?

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11. Suppose X and Y are random variables with μX = 10, σX = 3, μY = 15, and σY = 4. Given that X and Y are independent, what are the mean and standard deviation of the random variable X+Y? μX+Y = μX + μY σX+Y = √σ2X + σ2Y = = √9+16 = 25 = √25 = 5

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**You roll a die. If it comes up a 6, you win. $100**

You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 the second time, you win $50. If not, you lose. a. Create a probability model for the amount you will win at this game.

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**You roll a die. If it comes up a 6, you win. $100**

You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 the second time, you win $50. If not, you lose. a. Create a probability model for the amount you will win at this game. Winnings $100 $50 $0 Probability 1/6 (5/6)(1/6) 5/36 (5/6)(5/6) 25/36

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**You roll a die. If it comes up a 6, you win. $100**

You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 the second time, you win $50. If not, you lose. b. Find the expected amount you’ll win.

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**You roll a die. If it comes up a 6, you win. $100**

You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 the second time, you win $50. If not, you lose. b. Find the expected amount you’ll win.

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Law of Large Numbers Law of Large Numbers – The long run relative frequency of repeated independent trials gets closer and closer to the true relative frequency as the number of trials increases.

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**Binomial Distributions**

Binomial Distribution – the distribution of the count X successes in the binomial setting. B(n,p) where n is the number of observations and p is the probability of success

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**Binomial Distributions**

Use binompdf(n,p,X) to find the probability of a single value of X, such as P(X = 3). Use binomcdf(n,p,X)to find the probability of at most X successes for example P(X ≤ 3).

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**Binomial Distributions**

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**13. Pepsi is running a sales promotion in**

13. Pepsi is running a sales promotion in which 12% of all bottles have a “FREE” logo under the cap. What is the probability that you find two free cans in a 6-pack?

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**13. Pepsi is running a sales promotion in**

13. Pepsi is running a sales promotion in which 12% of all bottles have a “FREE” logo under the cap. What is the probability that you find two free cans in a 6-pack?

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**13. Pepsi is running a sales promotion in**

13. Pepsi is running a sales promotion in which 12% of all bottles have a “FREE” logo under the cap. What is the probability that you find two free cans in a 6-pack?

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**14. The National Association of Retailers**

14. The National Association of Retailers reports that 62% of all purchases are now made by credit card; you think this is true at your store as well. On a typical day you make 20 sales. a. Let X represent the number of customers who use a credit card on a typical day. What is the probability model for X?

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**14. The National Association of Retailers**

14. The National Association of Retailers reports that 62% of all purchases are now made by credit card; you think this is true at your store as well. On a typical day you make 20 sales. a. Let X represent the number of customers who use a credit card on a typical day. What is the probability model for X? The model is = B(20, .62) [i.e. B(n,p)] Please Explain Why

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**14. The National Association of Retailers**

14. The National Association of Retailers reports that 62% of all purchases are now made by credit card; you think this is true at your store as well. On a typical day you make 20 sales. b. Find the mean and standard deviation.

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**14. The National Association of Retailers**

14. The National Association of Retailers reports that 62% of all purchases are now made by credit card; you think this is true at your store as well. On a typical day you make 20 sales. b. Find the mean and standard deviation.

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**14. The National Association of Retailers**

14. The National Association of Retailers reports that 62% of all purchases are now made by credit card; you think this is true at your store as well. On a typical day you make 20 sales. c. What is the probability that on a typical day at least half of your customers use a credit card?

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**14. The National Association of Retailers**

14. The National Association of Retailers reports that 62% of all purchases are now made by credit card; you think this is true at your store as well. On a typical day you make 20 sales. c. What is the probability that on a typical day at least half of your customers use a credit card?

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**19. The volumes of soda in quart soda bottles**

19. The volumes of soda in quart soda bottles can be described by a Normal model with a mean of 32.3 oz and a standard deviation of 1.2 oz. What is the probability that a randomly selected bottle has a volume less than 32 oz?

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**P(x< 32) = P(z < [32-32.3]/1.2) =**

There are typos in the next slide: The z score calculation should read: P(x< 32) = P(z < [ ]/1.2) = P(z< -.25) = .4013 And Normalcdf(-E99, 32, 32.3, 1.2) = .4013

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**The volumes of soda in quart soda bottles**

The volumes of soda in quart soda bottles can be described by a Normal model N(32.3, 1.2) a mean of 32.3 oz and a standard deviation of 1.2 oz. What is the probability that the volume of a randomly selected bottle has a less than 32 oz?

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**20. A bank's loan officer rates applicants for. credit**

20. A bank's loan officer rates applicants for credit. The ratings can be described by a Normal model with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, what is the probability that the rating is between 200 and 275?

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**20. A bank's loan officer rates applicants for. credit**

20. A bank's loan officer rates applicants for credit. The ratings can be described by a Normal model with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, what is the probability that the rating is between 200 and 275?

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**Sampling Distributions**

Sampling distribution – the distribution of values taken by a statistic in all possible samples of the same size from the same population

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**Sampling Distributions of Sample Proportions**

Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of is modeled by a Normal model with mean and standard deviation

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**Assume that 12% of students at a university wear contact lenses**

Assume that 12% of students at a university wear contact lenses. We randomly pick 200 students. What is the mean of the proportion of students in this group who may wear contact lenses? What is the standard deviation of the proportion of students in this group who may wear contact lenses?

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**Assume that 12% of students at a university wear contact lenses**

Assume that 12% of students at a university wear contact lenses. We randomly pick 200 students. What is the mean of the proportion of students in this group who may wear contact lenses? What is the standard deviation of the proportion of students in this group who may wear contact lenses?

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**Sampling Distribution of a Sample Mean**

Suppose that x-bar is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ. Then the mean of the sampling distribution of xbar is μ (hence xbar is an unbiased indicator of μ) standard deviation of the sampling distribution of xbar is σ /√n.

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The scores of individual students on the ACT have a normal distribution with mean and standard deviation At Northside High, 76 seniors take the test. If the scores at this school have the same distribution as national scores, what are the mean and standard deviation of the distribution of sample means for these 76 students?

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The scores of individual students on the ACT have a normal distribution with mean and standard deviation At Northside High, 76 seniors take the test. If the scores at this school have the same distribution as national scores, what are the mean and standard deviation of the average (sample mean) for the 76 students?

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