Download Ch. 06 - Notes - Probability

Ch. 6 - Probability - Stahler - Statistics - Page 1
Essential Question(s): How do you calculate probability?
Simulate tossing a coin 10 times with your calculator. Record the number of heads and the number of tails.
In the short run, the behavior (heads or tails) is unpredictable. What if we flipped the coin 100 times? 100,000?
After many repetitions, a pattern will clearly emerge. This is the basis for the idea of probability.
Probability, the idea of it, is based on observation rather than theorizing. We could say the probability of getting
tails is 0.5 but we must run many trials to pin down a probability (you can imagine what would happen if we did
this an infinite number of times).
Random: a phenomenon is random iff individual outcomes are uncertain but there is nonetheless a regular
distribution of outcomes in a large number of repetitions
Probability: any outcome of a random phenomenon is the proportion of times the outcome would occur in a
very long series of repetitions. That is, probability is long-term relative frequency. % = f n
As we explore randomness remember:
a) you must have a long series of independent trials
b) we can estimate real-world probability only by observing many trials (we can get computers to help)
c) short runs only give rough estimates of a probability
Where do we use probability? Gambling (How do you play BlackJack?).
Where do probabilities come from?
1) Observed Data (long-run relative frequencies).
Ex: Observation of thousands of births has shown that about 51% of newborns are boys. You can use
this data to say that the probability of the next newborn being a boy is about .51.
2) Symmetry (equally likely outcomes)
Ex: Flipping a coin
3) Subjective Estimates
Ex: What is the probability you will get an A in this class? The use of probability is meaningful in this
situation, but you cannot gather data or list equally likely outcomes. You can make a subjective
judgment.
Assigning Probabilities: Equally likely outcomes
We must observe many outcomes to assign probabilities. There are some special circumstances where we are
willing to assume individual outcomes are equally likely. Flipping fair coins and rolling fair dice.
1
Ex: Think back to our Random Number table from chapter 5. Are the digits (0-9) equally like to appear? What
is the sample space S? Create a probability model (distribution/table) showcasing the probability of getting
each digit.
If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has
probability 1/k. (*Most random phenomena do not have equally likely outcomes)
P( A) =
# of outcomes in A # of outcomes in A
=
# of outcomes in S
k
We can assign probabilities in 2 ways.
1) Empirically (statistical) - probabilities assigned from our knowledge of numerous past events
Ex: Mendel discovered the probabilities of inheritance of a given trait from experiments on peas
without knowing about genes or DNA.
Ex: The probability that a voter chosen at random will vote Republican is 0.45.
2) Theoretically (classic) - probabilities from our understanding of the phenomenon and symmetries in the
problem
Ex: A 6-sided fair die: each side has the same chance of turning up
Ex:
Law of Large Numbers: as an experiment is repeated over and over, the empirical probability of an event
approaches the theoretical probability of the event. (tossing coins, rolling dice,...)
Ex: When Shaq was playing basketball, he would make about half of his free throws. Simulate 100 free throws
using your calculator. What will we assign for a make? Miss? Store this in list “Shaq.”
a) Looking at your list, what is the longest run of makes? Longest run of misses?
b) What percent of the 100 shots did SHAQ make? *Make a bar chart of LShaq and use the “trace” function.
Ex: Probability is a measure of how likely an event is to occur. Match the following probabilities with each
statement about an event. 0, 0.01, 0.3, 0.6, 0.99, 1
a) This event is impossible. It can never happen.
b) This event is certain. It will occur on every trial of the random phenomenon.
c) This event is unlikely, but it will occur once in awhile in a long sequence of trials.
d) This event will occur more often than not.
Ex: You read in a book on poker that the probability of being dealt a three of a kind in a five-card poker hand is
1/50. Explain in simple language what this means.
2
Ch. 6 - Probability - Stahler - Statistics - Page 3
Essential Question(s): How do you construct a probability model?
Flip a coin. What are the outcomes? What is the probability of each outcome? Show prob sim on calc (APPS).
We use models in previous chapters to describe linear relationships (scatterplots). A probability model
consists of the following.
a) The sample space S of a random phenomenon is the set of all possible outcomes
1) Each outcome is assigned a probability between 0 and 1.
2) The sum of all probabilities is 1
b) A way of assigning probabilities to events.
*An event is any outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the
sample space.
Ex: You roll two dice and record the sum. What is the sample space? Give examples of 3 different events.
Ex: Using the random table in chapter 5, let your pencil drop (graphite down) on the numbers. List the possible
outcomes for the sample space.
Ex: You are asking students at Berks Catholic if they are traveling during the holiday season. What is the
sample space?
Ex: Flip a fair coin then roll a standard 6-sided die. Illustrate the sample space using a tree diagram. How
many outcomes are there for flipping the coin? How many outcomes are there for rolling the die? How many
total outcomes are there?
Fundamental Counting Principle: If you can do one task in a number of ways and you can do another task b
number of ways, both tasks can be done in a x b number of ways.
Ex: You are getting dressed this Thursday morning and your power is out. You cannot see a thing. You have 7
pairs of underwear (one for each day of the week). You wanted to wear your Thanksgiving Holiday t-shirt (you
have a total of 5 t-shirts in your t-shirt drawer). You are wearing your only pair of jeans but you need to pick
your brown belt (you have either a brown or black belt). How many outfits exist? What is the probability you
picked the correct outfit? What is the probability you did not pick the correct outfit?
Ex: Describe the composition of a standard deck of cards?
Ex: Place 10 pieces of paper in a hat. The papers are numbered from 0-9. How many different ways can you
make a 3 digit number with replacement of the papers? How many ways can you make a 3 digit number
without replacement?
3
HW:
1) In each of the following situations, describe a sample space S for the random phenomenon. In some cases,
answers may vary.
a) A seed is planted in the ground. It either germinates or fails to grow.
b) A patient with athlete’s foot is given a new treatment to relieve the itch. The response variable is the length
of time that the patient waits for before relief.
c) A student enrolls in a statistics course and at the end of the semester receives a letter grade.
d) A basketball player shoots two free throws. You record the sequence of hits and misses.
e) A basketball player shoots four free throws. You record the number of baskets he makes.
2) For each of the following use a tree diagram or the fundamental counting principle to determine the number
of outcomes in the sample space. Then write the sample space using set notation (S={...}).
a) Toss 2 coins (one and then the other).
b) Toss 3 coins (one and then the other).
3) For each of the following, use the fundamental counting principle to determine the number of outcomes in
the sample space.
a) Suppose a county license tag has a four-digit number for identification. If any digit can occupy any of the
four positions, how many county license tags can you have?
b) If the county license tags described above do not allow for duplicate digits, how many county license tags
can you have?
c) Suppose the county license tags can have up to four digits. How many county license tags will this scheme
allow (repetition allowed)?
4) Suppose you pick a card from a standard deck. In how many ways can the selected card be
A red card?
A heart?
A queen and a heart?
A queen or a heart?
A queen that is not a heart?
4
5) What does it mean to be mutually exclusive? You might need a dictionary for this. Give an example.
What is a synonym for mutually exclusive? Draw a Venn diagram to illustrate your example.
6) What does it mean to be non-mutually exclusive? You might need a dictionary for this. Give an example.
What is a synonym for non-mutually exclusive? Draw a Venn diagram to illustrate your example.
5
Ch. 6 - Probability - Stahler - Statistics - Page 6
Essential Question(s): How do we calculate the probability of successive events?
Basic
1) Any probability is a number between 0 and 1 (inclusive).
0 ≤ 𝑃(𝐴) ≤ 1
2) The sum of all possible outcomes must be 1.
3) The probability that an event does not occur is 1 minus the probability that it does occur.
Ac is known as the complement. P(AC) = 1 - P(A)
also denoted: A’ or A
Not Basic
1) Multiplication Rule (compound events)
General:
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵|𝐴)
Independent: 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵)
2) Conditional Probability
P ( A B) =
P ( A and B)
P ( B)
*This comes from the formula above.
3) Addition Rule
General:
P(A or B) = P(A) + P(B) – P(A and B)
Mutually Exclusive: P(A or B) = P(A) + P(B)
We will begin our tour of probability looking at the multiplication rule. We use the multiplication rule when we
have compound events (events that occur in succession).
Ex: What is the probability of rolling a 6 and picking a face card from a standard deck?
Ex: What is the probability of picking a face card and then pick a 7 (without replacement).
The probability that two events will occur in succession is 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵|𝐴).
“P(B|A) is read: “The probability of B given A.”
Two events A and B are independent if knowing that one occurring does not change the probability that the
other occurs. If A and B are independent then 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵)
Ex: Explain why we can shorten the formula for independent events.
Ex: The last 4 digits of your phone number are random and independent. What is the probability the last 4
digits of your phone number are 5309?
6
Ex: Suppose you and your spouse are having a child. You both carry the genes for brown and blue eyes
(recessive). In order for your child to have blue eyes, you BOTH must pass on the blue eye gene. What is the
probability that you pass on the blue eye gene? Your spouse? Both of you?
Ex: Suppose that 64% of registered voters are Democrat. If you interviewed 2 voters chosen independently.
What is the probability the first is a Democrat and the second is not?
Ex: The first successful transatlantic telegraph cable was laid in 1866. The first telephone cable across the
Atlantic did not appear until 1956—the barrier was designing “repeaters,” amplifiers needed to boost the signal,
that could operate for years on the sea bottom. This first cable had 52 repeaters. The copper cable, laid in
1983 and retired in 1994, had 662 repeaters. The first fiber optic cable was laid in 1988 and has 109 repeaters.
There are now more than 400,000 miles of undersea cable, with more being laid every year to handle the flood
of internet traffic.
Repeaters in undersea cables must be very reliable. To see why, suppose that each repeater has probability
0.999 of functioning without failure for 25 years. Repeaters fail independently of each other. (This assumption
means that there are no “common causes” such as earthquakes that would affect several repeaters at
once.) Denote Ai the event that the ith repeater operates successfully for 25 years.
What is the probability that one repeater lasts for 25 years? Two repeaters? Three repeaters? Ten repeaters?
The last copper transatlantic cable had 662 repeaters. What is the probability that all 662 repeaters work for 25
years? What is the probability that at least one fails?
HW:
1) A general can plan a campaign to fight one major battle or three small battles. He believes that he has
probability 0.6 of winning the large battle and probability 0.8 of winning each of the small battles. Victories or
defeats in the small battles are independent. The general must win either the large battle or all three small
battles to win the campaign. Which strategy should he choose?
2) An automobile manufacturer buys computer chips from a supplier. The supplier sends a shipment containing
5% defective chips. Each chip chosen from this shipment has probability 0.05 of being defective, and each
automobile uses 12 chips selected independently. What is the probability that all 12 chips in a car will
work properly?
7
3) An athlete suspected of having used steroids is given two tests that operate independently of each other.
Test A has probability 0.9 of being positive if steroids have been used. Test B has probability 0.8 of being
positive if steroids have been used. What is the probability that neither test is positive if steroids have been
used?
4) Choose a new car or light truck at random and note its color. Here are the probabilities of the most popular
colors for vehicles made in North America in 2000.
Color
Silver
White
Black
Dark Green
Dark Blue
Medium Red
Probability
0.176
0.172
0.113
0.089
0.088
0.067
a) What is the probability that the vehicle you choose has any color other than the six listed?
b) What is the probability that a randomly chosen vehicle is either silver or white?
c) Choose two vehicles at random. What is the probability that both are silver or both are white?
5) A roulette wheel has 38 slots numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the
others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the
wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any
slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors.
a) What is the probability that the ball will land in any one slot?
b) If you bet on “red,” you win if the ball lands in a red slot. What is the probability of winning?
c) The slot numbers are laid out on a board on which gamblers place their bets. One column of numbers on the
board contains all multiples of 3, that is, 3, 6, 9, . . . , 36.You place a “column bet” that wins if any of these
numbers comes up. What is your probability of winning?
8
Ch. 6 - Probability - Stahler - Statistics - Page 9
Essential Question(s): How do we calculate conditional probability? How do we determine if events are indep?
Rv: You take 10 free-throws and you make 7 of them.
a) What is the probability you make a free-throw?
Video
b) What are the odds you make a free-throw?
Rv: Solve 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵|𝐴) for 𝑃(𝐵|𝐴)
Conditional Probability: The probability of an event occurring, given that another event has already occurred.
The conditional probability of event B occurring, given that event A has occurred, is denoted by P(B|A) and is
read as “probability of B, given A.”
Ex: As a class, complete the table below. You should be able to place yourself in one of the 4 categories.
Brown Eyes
Not Brown Eyes
Total
Dark Hair
Not Dark Hair
Total
Calculate the following:
a) P(Brown Eyes)
c) P(Not Dark Hair|Brown Eyes)
b) P(Not Brown Eyes|Dark Hair)
d) P(Brown Eyes and Not Dark Hair)
*Show this 2 different ways.
Are the events “Brown Eyes” and “Dark Hair” independent events? How do you know?
How to Determine if Two Events (A and B) are Independent (mathematically)
1) Find the probability of one of the events: P(A)
2) Find the probability of the same event given that the other event already occurred: P(A|B)
3) If the values from Step 1 and 2 above are the equal, they are independent.
If the values from Step 1 and 2 above are not equal, they are dependent.
Ex: Are the events “Rolling an Even #” and “Flipping Tails” independent? Show mathematically.
Ex: Are the events “Pulling a Black Queen” and then “Pulling a Diamond” (w/o replacement) independent?
Show mathematically.
9
HW:
1) The table to the right shows the results of a survey in which
146 families were asked if they own a computer and if they will
be taking a summer vacation this year.
a) Find the probability that randomly selected family is not taking
a summer vacation this year.
b) Find the probability a family is taking a summer vacation this year given that they own a computer.
c) Find the probability a family owns a computer and they are not taking a summer vacation.
d) Are the events of owning a computer and taking a summer vacation this year indep. or dep.? Explain
mathematically.
2) In a jury selection pool, 65% of the people are female. Of these 65%, one out of four works in a health field.
a) Find the probability that a randomly selected person from the jury pool is female and works in a
health field
b) Find the probability that a person from the jury pool is female and does not work in a health field.
3) The probability that a particular knee surgery is successful is 0.85.
a) Find the probability that three knee surgeries are successful. Are these events dep. or indep.?
b) Find the probability that none of the surgeries are successful.
c) Find the probability that at least one of the three surgeries is successful.
4) Choose an employed person at random. Let A be the event that the person chosen is a women, and B the
event that the person holds a managerial or professional job. Government data tell us that P(A)=0.46 and the
probability of managerial and professional jobs among women is P(B|A)=0.32. Find the probability that a
randomly chosen employed person is a woman holding a managerial or professional position.
5) Functional Robotics Corporation buys electrical controllers from a Japanese supplier. The company’s
treasurer thinks that there is probability 0.4 that the dollar will fall in value against the Japanese yen in the next
month. The treasurer also believes that if the dollar falls there is probability 0.8 that the supplier will demand
renegotiation of the contract. What probability has the treasurer assigned to the event that the dollar falls and
the supplier demands renegotiation?
10
Ch. 6 - Probability - Stahler - Statistics - Page 11
Essential Question(s): How do we calculate the probability of mutually exclusive events?
What does it mean to be mutually exclusive (disjoint)? Give an example.
What does it mean to be not mutually exclusive (joint)? Give an example.
*Disjoint events are not independent events.
Ex: Disjoint events cannot both occur at the same time. If one happens, the other does not – guaranteed.
Independent events must be able to occur at the same time. If one happens, it has no influence on the other
whatsoever. The occurrence of one provides no information about the other.
I draw a card from a deck. You have to bet on whether or not it’s red. You have a 50-50 chance of being right.
Now suppose that before you bet, I peek at the card and tell you it’s a spade. Does that help you place your
bet? Of course! Note that the events “red” and “spade” are disjoint – there are no red spades. Because of that,
knowing the card is a spade is a dead giveaway; it tells you for sure that the card can’t be red. The probability
the card is red has changed from 1/2 to 0. These two events are NOT independent.
OK, let’s play again. This time I peek at the card and tell you it’s an ace. Does that help you place your bet?
Before you knew this, the probability the card is red was 26/52 = 1/2. Knowing it’s an ace, the probability it’s
red is 2/4 = 1/2. No help whatsoever – the probability has not changed. These two events ARE independent
(and not disjoint). P(red | ace) = P(red) — that’s the very definition of independence:
the occurrence of “ace” has no effect on the probability of “red”.
The Addition Rule:
General:
Mutually Exclusive:
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝑃) – 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
Ex: Explain why it is not necessary to subtract P(A and B) if the events are mutually exclusive (disjoint).
Ex: Given the following probability model, (marital status of women 25-34) answer the questions below.
Marital Status:
Probability:
Never Married
Married
Widowed
Divorced
0.298
0.622
0.005
0.075
a) Do the probabilities add up to 1?
b) Show 2 different ways to find the % of females sampled that are not married.
c) Being “Married” and “Divorced” is a disjoint event. What is the probability that women in the age group are
either married or divorced?
11
Ex: You are dealt one card from a standard deck. Find the following probabilities.
a) P(King or Queen)
b) P(King or Red)
c) P(Black or Face)
d) P(King and Queen)
f) P(Not Spade)
e) P(Ace and Spade)
Set Notation
{𝐴 È𝐵} read “A union B” is the set of all outcomes that are either in A or in B.
Ex: King or Queen when pulling 1 card. (draw a picture)
{𝐴 ∩ 𝐵} read “A intersect B” is the set of all outcomes that are in both A and B.
Ex: Red and Jack when pulling 1 card. (draw a picture)
*If two events are disjoint then {𝐴 ∩ 𝐵} = ⊘
⊘ is used to represent the empty set (no outcomes).
Ex: Rewrite the equations for the addition rule using set notation.
Not Mutually Exclusive
Mutually Exclusive
Venn Diagram: use of circle to represent relationships between sets
Ex: Mutually Exclusive
Ex: Not Mutually Exclusive
Benford’s Law Video
First Digit:
1
2
3
4
5
6
7
8
9
Probability
0.301
0.176
0.125
0.097
0.079
0.067
0.058
0.051
0.046
HW: 1) All human blood can be typed as one of O, A, B, or AB, but the distribution of the types varies a bit with
race. Here is the distribution of the blood type of a randomly chosen Bulgarian.
Blood Type
O
A
B
AB
Probability
32%
44%
15%
?%
a) What is the probability of type AB blood? Why?
b) Grigor has blood type B. He can safely receive blood transfusions from people with blood type O and B.
What is the probability that a randomly chosen Bulgarian can donate blood to Grigor?
12
2) Suppose you have a Twister spinner with 4 different sections. Determine the theoretical probability of each
section. Under APPS in your calculator, you will find Prob Sim. Use the Spin Spinner to simulate 12 spins (with
4 sections). Make a table of your results and compare the experimental probabilities with the theoretical
probabilities.
3) Using the probabilities from Benford’s Law, answer the following.
First Digit:
1
2
3
4
5
6
7
8
9
Probability
0.301
0.176
0.125
0.097
0.079
0.067
0.058
0.051
0.046
Consider the following:
P(A)=P(first digit is 1) = 0.301
P(B)=P(first digit is 6 or greater) = 0.222
P(C)=P(first digit is odd) = 0.609
We will define event D to be (first digit is less than 4). Using the union and intersection notation, find the
following probabilities.
c) P(Dc)
b) 𝑃(𝐵 ∪ 𝐷)
a) P(D)
d) 𝑃(𝐶 ∩ 𝐷)
e) 𝑃(𝐵 ∩ 𝐷)
f) 𝑃(𝐵 ∪ 𝐶)
4) Two psychologists surveyed 478 children in grades 4, 5, and 6 in elementary schools in Michigan. They
stratified their sample, drawing roughly ⅓ from rural, ⅓ from suburban, and ⅓ from urban schools. Among
other questions, they asked the students whether their primary goal was to get good grades, to be popular, or
to be good at sports. One question of interest was whether boys and girls at this age had similar goals. Here is
a contingency table giving the counts of the students by the goals and sex:
Grades
Popular
Sports
Total
Boy
117
50
60
227
Girl
130
91
30
251
Total
247
141
90
478
a) What is the probability that a selected student wants to excel at sports given that we have selected a girl?
Write this conditional probability using probability notation. Find the answer using two different methods.
b) What is the probability that a student is a boy given that they want to be popular? Write this conditional
probability using probability notation. Find the answer using two different methods.
c) What is the probability that a student has a primary goal of grades or being popular?
d) What is the probability that a student is a boy or has a primary goal of sports?
13
5) A check of dorm rooms on a large college campus revealed that 38% had refrigerators, 52% had TVs, and
21% had both a TV and a refrigerator. What’s the probability that a randomly selected dorm room has
a) a TV but no refrigerator?
b) a TV or a Refrigerator, but not both?
c) neither a TV nor a refrigerator?
6) Fifty-six percent of all American workers have a workplace retirement plan, 68% have health insurance, and
49% have both benefits. We select a worker at random.
a) What is the probability that he has neither employer-sponsored health insurance nor a retirement plan?
b) What’s the probability he has health insurance if he has a retirement plan?
7) Call a household prosperous if its income exceeds $100,000. Call the household educated if the
householder completed college. Select an American household at random, and let A be the event that the
selected household is prosperous and B the event that it is educated. According to the Census Bureau, P(A) =
0.134, P(B) = 0.254, and the joint probability that a household is both prosperous and educated is P(A and B) =
0.080. What is the probability P(A or B) that the household selected is either prosperous or educated?
8) Common sources of caffeine are coffee, tea, and cola drinks.
Suppose that
and also that
55% of adults drink coffee
15% drink both coffee and tea
25% of adults drink tea
5% drink all three beverages
45% of adults drink cola
25% drink both coffee and cola
5% drink only tea
Draw a Venn diagram marked with this information. Use it along with the addition rules to answer the following
questions.
a) What percent of adults drink only cola?
b) What percent drink none of these beverages?
14
9) Police report that 78% of drivers stopped on suspicion of drunk driving are given a breath test, 36% a blood
test, and 22% both tests.
Draw a Venn diagram to illustrate the given probabilities. What is the value of the region outside of the
diagram?
a) What is the probability that a randomly selected DWI suspect is given a test?
b) What is the probability that the suspect gets either a blood test or a breath test but NOT both?
c) What is the probability that the suspect gets neither test?
10) A complex electronic device contains three components, A, B, and C. The probabilities of failure for each
component in any one year are 0.01, 0.03, and 0.04, respectively. If any one component fails, the device will
fail. If the components fail independently of one another, what is the probability that the device will not fail in
one year (2012)?
a) Less than 0.01
b) 0.078
c) 0.080
d) 0.922
e) Greater than 0.99
11) The probability that a new microwave oven will stop working in less than 2 years is 0.05. The probability
that a new microwave oven is damaged during delivery and stops working in less than 2 years is 0.04. The
probability that a new microwave is damaged during delivery is 0.10. Given that a new microwave oven is
damaged during delivery, what is the probability that it stops working in less than 2 years (2012)?
a) 0.05
b) 0.06
c) 0.10
d) 0.40
e) 0.50
Make sure you can interpret the following Venn diagrams (www.xkcd.com).
Euler Diagram
15