Date post: | 08-Jan-2018 |
Category: |
Documents |
Upload: | emerald-lang |
View: | 387 times |
Download: | 10 times |
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 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
.1.2
.3
.4
P(x)
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:
Mean, Variance, & Standard Deviation
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.
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 0.004 0.000
1 0.435 0.435
2 0.355 0.710
3 0.206 0.618
1. Find the mean. Do this by creating an x*P(x) column and adding up its values.
2. 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 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
1. Find the mean. Do this by creating an x*P(x) column and adding up its values.
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
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 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
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: $23,000St. 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.
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
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:
• 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,…}
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
[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
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 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.
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!
-μx
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.