Chapter 4: Discrete Probability Distributions

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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 DistributionsA 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.

x P(x)0 0.0041 0.4352 0.3553 0.206

number ofvehicles

Conditions of a prob. distribution• Each probability must be between

0 and 1, inclusive. • The sum of all probabilities is 1.

0 1 2 30

Number of Vehicles

x0 1 2 3

Identifying Distributions

• Which of the following is a discrete random distribution? Explain.

x p(x) x p(x) x p(x)

0 0.23 0 0.4 0 0.1

1 0.57 1 0.5 1 0.2

2 1.1 2 0.3 2 0.4

3 -0.9 3 0.1 3 0.3

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:

Find the mean, variance, and standard deviation of:

x P(x) x*P(x)

0 0.004

1 0.435

2 0.355

3 0.206

1. Find the mean. Do this by creating an x*P(x) column and adding up its values.

x P(x) x*P(x) (x–μ)²*P(x)

0 0.004 0.000

1 0.435 0.435

2 0.355 0.710

3 0.206 0.618

2. Find variance. Do this by creating a (x–μ)²*P(x) column and adding up its values.

Mean: 1.763 Variance:

x P(x) x*P(x) (x–μ)²*P(x)

0 0.004 0.000 0.012

1 0.435 0.435 0.253

2 0.355 0.710 0.020

3 0.206 0.618 0.315

2. Find variance. Do this by creating a (x–μ)²*P(x) column and adding up its values.

3. Find standard deviation by taking the square root of variance

Mean: 1.763 Variance: .6 Standard Deviation: .775

Siblings Classwork1. Fill in the table with the

number of siblings your classmates have

2. Draw the corresponding probability histogram

3. Calculate the mean, variance, and standard deviation.

X Freq. P(X) x*P(x) (x–μ)²*P(x) 0

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

• 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 0 ½ 0 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 = $4.14St. Dev = $5.44

• 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.

• 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: $23,000St. Dev: $21,000

• 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.

• 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.

X P(x) X*P(x)

70,000 .18 12600 332820000

50,000 .12 6000 63480000

20,000 .42 8400 20580000

0 .28 0 204120000

Mean: $27,000St. Dev: $24919.87

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:

• 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.

More Practice• 60% of Americans wear either glasses or contacts. You select

at random four Americans. Find the following probabilities:

1. Exactly three people wears glasses or contacts.

2. Less than three people wears glasses or contacts.

3. 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,…}

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

[2nd ][vars][binomcdf]

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:

1. Exactly four students earn over $400 a month.

2. Less than four students earn over $400 a month.

3. 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

1. Fewer than five people carpool.

2. More than seven people carpool.

3. 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

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 1You pick boxes at random, where one in six have a prize. Find the probability that you…

a) Win your first prize on the 4th purchase

b) Win your first prize within your first three purchases

Geometric Practice 2A 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…

a) The fourth person will be the first person to make a purchase.

b) The first or the fourth person will be the first person to make a purchase.

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

• 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!

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…

(a) 3 earthquakes occur in a year.

(b) less than 8 earthquakes occur in a year.

(c) more than 5 earthquakes occur in a year.

(d) 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 …

(a) …two children.

(b) … more than four children.

(c) … fewer than five children.

(d) No more than six children.

Chapter 4: Discrete Probability Distributions

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