Chapter 4: Discrete Probability Distributions Lesson 4.1: Probability Distributions (part 1)

Random Variables A random variable represents the numerical value of the outcome of a probability experiment A random variable is discrete if the number of possible outcomes is finite or countable. Example: The number of people in a car A random variable is continuous if it can take on any value within an interval. Example: The gallons of gas bought in a week

Discrete Probability Distributions A discrete probability distribution lists the possible values of the random variable, with its probability. Example: A survey asks a sample of families how many vehicles each owns. number of vehicles 1 2 3 .1 .2 .3 .4 P(x) Number of Vehicles x Conditions of a prob. distribution Each probability must be between 0 and 1, inclusive. The sum of all probabilities is 1.

Identifying Distributions Which of the following is a discrete random distribution? Explain. x p(x) 0.23 0.4 0.1 1 0.57 0.5 0.2 2 1.1 0.3 3 -0.9

Mean, Variance, & Standard Deviation The mean (expected value) of a discrete probability distribution is: The variance of a discrete probability distribution is: The standard deviation of a discrete probability distribution is:

Mean, Variance, & Standard Deviation Find the mean, variance, and standard deviation of: x P(x) x*P(x) 0.004 1 0.435 2 0.355 3 0.206 Find the mean. Do this by creating an x*P(x) column and adding up its values. Mean:

Mean, Variance, & Standard Deviation Find the mean, variance, and standard deviation of: x P(x) x*P(x) (x–μ)²*P(x) 0.004 0.000 1 0.435 2 0.355 0.710 3 0.206 0.618 Find the mean. Do this by creating an x*P(x) column and adding up its values. Find variance. Do this by creating a (x–μ)²*P(x) column and adding up its values. Mean: 1.763 Variance:

Mean, Variance, & Standard Deviation Find the mean, variance, and standard deviation of: x P(x) x*P(x) (x–μ)²*P(x) 0.004 0.000 0.012 1 0.435 0.253 2 0.355 0.710 0.020 3 0.206 0.618 0.315 Find the mean. Do this by creating an x*P(x) column and adding up its values. Find variance. Do this by creating a (x–μ)²*P(x) column and adding up its values. Find standard deviation by taking the square root of variance Mean: 1.763 Variance: .6 Standard Deviation: .775

Siblings Classwork Fill in the table with the number of siblings your classmates have Draw the corresponding probability histogram Calculate the mean, variance, and standard deviation. X Freq. P(X) x*P(x) (x–μ)²*P(x) 1 2 3 4 5 6 P(x) x

Chapter 4: Discrete Probability Distributions Lesson 4.1: Probability Distributions (part 2)

Creating Probability Distributions 1 Construct the probability distribution and compute the expected value (mean) and standard deviation You draw a card from a deck. If you get a red card you win nothing. If you get a spade, you win $5. For any club you win $10 plus an extra $20 if you pull the ace of clubs

Creating Probability Distributions 1 Construct the probability distribution and compute the expected value (mean) and standard deviation You draw a card from a deck. If you get a red card you win nothing. If you get a spade, you win $5. For any club you win $10 plus an extra $20 if you pull the ace of clubs Outcome x P(x) X*P(x) red ½ 8.570 spade 5 ¼ 1.25 0.185 Club (not ace) 10 12/52 2.308 7.925 Ace of Clubs 30 1/52 0.577 12.86 Mean = $4.14 St. Dev = $5.44

Creating Probability Distributions 2 Construct the probability distribution and compute the expected profit and standard deviation Bob purchases a house for $120,000 and plans to flip it. He spends $50,000 in repairs. He estimates he has a 20% chance of selling it for $160,000, a 50% chance of selling it for $190,000, and a 30% chance of selling it for $220,000.

Creating Probability Distributions 2 Construct the probability distribution and compute the expected profit and standard deviation Bob purchases a house for $120,000 and plans to flip it. He spends $50,000 in repairs. He estimates he has a 20% chance of selling it for $160,000, a 50% chance of selling it for $190,000, and a 30% chance of selling it for $220,000. X P(x) X*P(x) -10,000 .2 -2,000 217800000 20,000 .5 10,000 4500000 50,000 .3 15,000 218700000 Mean: $23,000 St. Dev: $21,000

Creating Probability Distributions 3 Construct the probability distribution and compute the expected profit and standard deviation A small software company bids on two contracts. It anticipates a profit of $50,000 if it gets the larger contract and a profit of $20,000 if it gets the smaller contract. The company estimates there is a 30% chance it will get the larger contract and a 60% chance it will get the smaller contract.

Creating Probability Distributions 3 Construct the probability distribution and compute the expected profit and standard deviation A small software company bids on two contracts. It anticipates a profit of $50,000 if it gets the larger contract and a profit of $20,000 if it gets the smaller contract. The company estimates there is a 30% chance it will get the larger contract and a 60% chance it will get the smaller contract. Mean: $27,000 St. Dev: $24919.87 X P(x) X*P(x) 70,000 .18 12600 332820000 50,000 .12 6000 63480000 20,000 .42 8400 20580000 .28 204120000

Chapter 4: Discrete Probability Distributions Lesson 4.2: Binomial Distributions (Part 1)

The Binomial Distribution Must satisfy the following conditions: There is a fixed number of independent trials There are two possible outcomes for each trial The probability of success is the same for each trial “x” represents the number of successful trials Binomial Experiments: You roll a die 10 times and record the number of 6s. What is the probability you rolled three 6s? 34% of people are blue eyed. You survey 86 people and record how many blue eyed people there are. What is the probability you pick at least 20 blue eyed people?

Binomial Notation S = “Success” F = “Failure” p = probability of success Q = probability of failure n = the number of trials x = the number of success in n trials Example: You pick 4 cards from a standard deck of cards WITH replacement. What is the probability that you pick exactly 3 aces. S = Ace F = Not an Ace p = 1/13 q = 12/13 n = 4 x = 3

The Binomial Formula In a binomial experiment the probability of exactly x success in n trials is: 𝑃 𝑥 = 𝑛 𝐶 𝑥 𝑝 𝑥 𝑞 𝑛−𝑥 Example: A multiple choice test has 5 questions each of which has 4 choices, one of which is correct. You want to know the probability that you guess exactly 3 questions correctly. 𝑃 3 = 5 𝐶 3 (.25) 3 (.75) 2 =0.0879

More Practice 60% of Americans wear either glasses or contacts. You select at random four Americans. Find the following probabilities: Exactly three people wears glasses or contacts. Less than three people wears glasses or contacts. At least three people wears glasses or contacts.

Beware of Wording “less than 3” {0,1,2} “at least 3” {3,4,5,….} “at most 3” {0,1,2,3} “more than 3” {4,5,6,…}

Chapter 4: Discrete Probability Distributions Lesson 4.2: Binomial Distributions (Part 2)

Binomial on the Calculator Use your calculator! [2nd ][vars][binomcdf] binompdf (n,p,x) This is the probability of exactly x successes from n trials binomcdf (n,p,x) This is the probability of 0 through x successes from n trials

More Binomial Practice 1 28% of college students earn over $400 a month. You select at random ten college students. Find the following probabilities: Exactly four students earn over $400 a month. Less than four students earn over $400 a month. At least four students earn over $400 a month.

More Binomial Practice 2 About 10% of workers in the US commute to their jobs by carpooling. You randomly select 20 workers. Find the following probabilities Fewer than five people carpool. More than seven people carpool. Exactly three people carpool.

Mean and Variance Mean: µ = np Variance: σ² = npq Standard deviation: σ = SQRT(npq) Example: 12% of Americans are left handed. If you surveyed 70 people, what is the mean, variance, and standard deviation of left handed people? Mean = np = 70*.12 = 8.4 Variance = 70*.12*.88 = 7.392 Standard Deviation = SQRT(7.392) = 2.719

Chapter 4: Discrete Probability Distributions Lesson 4.3: More Discrete Probability Distributions (Part 1)

The Geometric Distribution Must satisfy the following conditions: A trial is repeated until a success occurs Repeated trials are independent of each other The probability of success is the same for each trial “x” represents the number of the trial in which the first success occurs Geometric Examples: 16% of cars are white, what is the probability the first white car you see is the fourth car that passes you? 23% of students at DHS are seniors. What is the probability the third person you survey is your first senior? P(x) = (q)x – 1p

Geometric on the Calculator [2nd ][vars][geometpdf] [2nd ][vars][geometcdf] geometpdf (p,x) This is the probability that the first success will occur on trial number x, where p is the probability of success for a single trial. geometcdf (p,x) This is the probability that the first success will occur between trial 1 through (and including) trial x.

Geometric Practice 1 You pick boxes at random, where one in six have a prize. Find the probability that you… Win your first prize on the 4th purchase Win your first prize within your first three purchases

Geometric Practice 2 A marketing study has found that the probability that a person who enters a particular store will make a purchase is 0.32. Find the probability that… The fourth person will be the first person to make a purchase. The first or the fourth person will be the first person to make a purchase.

Chapter 4: Discrete Probability Distributions Lesson 4.3: More Discrete Probability Distributions (Part 2)

The Poisson Distribution Must satisfy the following conditions: The experiment counts the number of times (x) an event occurs in some interval (time, area, etc.) The probability of the event occurring is the same for each interval The number of occurrences in each interval are independent of each other Poisson Examples: Each year there are an average of 6.8 shark attacks. What is the probability this year there will be 9 shark attacks? A textbook has an average of 0.7 typos per page. You randomly select one page, what is the probability there is more than 2 typos? P(x)= μ e x! -μ x

Poisson on the Calculator [2nd ][vars][poissonpdf] [2nd ][vars][poissoncdf] poissonpdf (μ,x) This is the probability that x occurrences of an event will occur over a specified interval of time, area, or volume. poissoncdf (μ,x) This is the probability that 0 through x occurrences of an event will occur over a specified interval of time, area, or volume.

Poisson Practice 1 California experiences on average 6.8 earthquakes of magnitude of 5 or higher every year. Find the probability that… 3 earthquakes occur in a year. less than 8 earthquakes occur in a year. more than 5 earthquakes occur in a year. at least 5 earthquakes occur in a year.

Poisson Practice 2 The average number of children per family in the United States is 1.86. You randomly select a family in the US, find the probability that they have … …two children. … more than four children. … fewer than five children. No more than six children.