5.1 Randomness and Probability

Name ______

Chapter 5 Learning Objectives / Section / Related Example
on Page(s) / Relevant
Chapter Review Exercise(s) / Can I do this?
Interpret probability as a long-run relative frequency. / 5.1 / 291, 292, 293, 294, 295 / R5.1
Use simulation to model chance behavior. / 5.1 / 296, 297 / R5.2
Determine a probability model for a chance process. / 5.2 / 306 / R5.3, R5.10
Use basic probability rules, including the complement rule and the addition rule for mutually exclusive events. / 5.2 / 308 / R5.4, R5.10
Use a two-way table or Venn diagram to model a chance process and calculate probabilities involving two events. / 5.2 / 309, 312, 313 / R5.4, R5.5, R5.7, R5.8
Use the general addition rule to calculate probabilities. / 5.2 / 313 / R5.4, R5.5
Calculate and interpret conditional probabilities. / 5.3 / 318, 320 / R5.6, R5.8
Use the general multiplication rule to calculate probabilities. / 5.3 / 322 / R5.6
Use tree diagrams to model a chance process and calculate probabilities involving two or more events. / 5.3 / 322, 324, 325 / R5.6
Determine whether two events are independent. / 5.3 / 327 / R5.7, R5.8
When appropriate, use the multiplication rule for independent events to compute probabilities. / 5.3 / 329, 331 / R5.9, R5.10

5.1 Randomness and Probability

Whose Phone is this?

Three friends joined Tom at his house to study for their next AP Stats test. They decided to put their (identical) phones on the kitchen table to avoid distractions for one hour while they go study in another room. After an hour, they returned to the kitchen and discovered that Tom’s little brother had moved the phones and stacked them one on top of another.They each grabbed a phone and checked to see whether they had selected their own, but found that none of the four had their own phone. How likely is it that none of the four students ended up with the correct phone?

Read 289-292

What is the law of large numbers?

How do you interpret a probability?

The probability of getting a sum of 11 when rolling two dice is 1/18. Interpret this value.

Imagine you are flipping a coin. Write down the results of 50 imaginary flips (e.g. HTTHT…):

Read 293-295

What are some myths about randomness?

Read 295, 296-297

What is the purpose of simulation?

The four steps for performing a simulation:

Do you need to do the four steps for each simulation?

Example: Sweet situation with my sweetener packets

Each morning as I make coffee, I grab 6 sweetener packets at random. The packets have one end that I prefer to tear open, rather than the other end—essentially they are “right side up” as I like to open them. One morning recently, all 6 packets were right side up as I grabbed them. Is this evidence that I’m just that good at selecting packets before having my morning dose of caffeine, or could this have occurred simply by chance? Assuming that 42% of the packets are right side up, and the way one packet is turned doesn’t depend on how any other packets are oriented, how likely is it for me to grab 6 or more packets turned right side up?

HW #35 page 300 (1, 3, 7, 9, 11)

5.1 Simulations

Read 296-297

What are some common errors when using a table of random digits?

Stratified Sampling: Suppose I want to choose a simple random sample of size 6 from a group of 60 boys and 30 girls. To do this, I write each person’s name on an equally sized piece of paper and mix them up in a large grocery bag. Just as I am about to select the first name, a thoughtful student suggests that I should stratify by class. I agree, and we decide it would be appropriate to select 4 boys and 2 girls. However, since I already mixed up the names, I don’t want to have separate them all again. Instead, I will select names one at a time from the bag until I get 4 boys and 2 girls. This means, however, that I may need to select more than 6 names (e.g. I may get more than 2 girls before I get the 4 boys). Design and carry out a simulation using Table D to estimate the probability that you must draw 9 or more names to get 4 boys and 2 girls.

HW #37 page 300 (15, 17, 19, 23, 25*, 27*, 29*, 38) *Do at least 10 trials

5.2 Probability Rules

Read 305-307

What is a sample space? What is a probability model?

What is an event?

Imagine spinning this spinner three times. Describe the probability

model for this chance process and use it to find the probability

of stopping on the black portion at least 1 time in three spins.

Read 307-309

What does it mean if two events are mutually exclusive (also called disjoint events)?

AP Statistics Scores: Randomly select a student who took the 2015 AP Statistics exam and record the student’s score. Here is the probability model:

Score / 1 / 2 / 3 / 4 / 5
Probability / 0.238 / 0.189 / 0.252 / 0.189 / 0.132

(a) Show that this is a legitimate probability model.

(b) Find the probability that the chosen student scored 3 or better.

HW #38: page 300 (31–36), page 314 (39, 41, 43, 45, 47)

5.2 Two-Way Tables

Read 309-311

What is the general addition rule?

Is it on the formula sheet?

What if the events are mutually exclusive?

College credit: The 120 students who took AP Stats at FHN one year could pay for UMSL credit for the class (or not) and could sign up to take the AP exam to try to earn college credit that way (or not). Overall, 32 students signed up for UMSL credit, 56 took the AP exam, and 8 students both signed up for UMSL credit and took the AP exam.

(a) Create a two-way table that displays the data.

Suppose we choose a student from among those AP Stats classes at random. Find the probability that the student

(b) signed up for UMSL credit.

(d) signed up for UMSL credit or took the AP exam.

Read 311-314

Movie Princesses: At one point, among the movie princesses from a particular studio, 25% had blonde hair, 25% had blue eyes, and 10% had both. Suppose we randomly select a princess from this studio at the time in question.

(a) Make a two-way table that displays the sample space of this chance process.

(b) Construct a Venn diagram to represent the outcomes of this chance process.

(d) Find the probability that the princess has neither blonde hair nor blue eyes.

(e) Find the probability the princess has blonde hair only.

HW #39page 314 (49–55 odd)

5.3 Conditional Probability and Independence

Read 318-321

What is a conditional probability?

50 Cent Drinks!

In 2014, Mobil On The Run Convenience Stores offered 50 cent beverages the day following a Cardinals game in which the team scored 6 or more runs. In the 2014 season, the Cards won 90 of their 162 regular season games and On The Run stores offered 50 cent beverages after 36 of those 162 games. In 35 games that season, On The Run stores offered 50 cent drinks following a Cardinals win.Let W = win and D = 50 cent drinks. Choose a 2014 Cards regular season game at random.

(a)Summarize these data in a two-way table.

(b)Find P(W or D)

(c)Find P(W | D)

(d)Find P(D | W)

Read 326-328

How can you tell if two events are independent?

More from the example 50 Cent Drinks!

(e) Are the events D and W independent? Explain.

Read 321-326

What is the general multiplication rule?

Is it on the formula sheet?

When is it better to use a tree diagram than a two-way table?

Headed to class: Because of the 2015 construction work on Willott Road, I travel to the night class I teach at SCC via a different route. Suppose there is a 0.2 probability I take 364, a 0.3 probability I cut through the Country Creek, and a 0.5 probability I take Sutter’s Mill. My goal is to get to SCC in less than 20 minutes. Suppose that the probability that I make it in under 20 minutes is 0.5 if I take 364, 0.6 if I go through Country Creek, and 0.7 if I take Sutter’s Mill. On a randomly chosen evening, what is the probability that I can make the trip in less than 20 minutes?

Read 328-331

What is the multiplication rule for independent events?

Is it on the formula sheet?

How is it related to the general multiplication rule?

What’s the difference between “mutually exclusive” and “independent”?

When are mutually exclusive events complementary events?

Create your own example of two events that are…

…Mutually exclusive (disjoint)

…Independent

…Neither disjoint nor independent

…Complements

My examples for you:

One trial: Rolling one die...

Two trials:

Bowl a 300 game: In bowling, a perfect game is when a bowler gets 12 strikes in a row. Suppose that I get a strike about 40% of the time. So, to bowl a 300 game, I would need to have 12 strikes in a row. What is the probability that I bowl 12 strikes in a row, assuming my performance in one frame is independent of my performance in other frames?

Suppose that at a very early point in pregnancy, there is about a 60% chance that an over-the-counter pregnancy test (OTCPT) will have a“false negative” result. Among 20 women with normal pregnancies at that point, what is the probability that there will be at least one false negative?

Weather Conditions: On a recent day, a forecast gave a 60% chance of rain in St. Charles and a 60% chance of rain in Maryland Heights. What is the probability it will rain in both locations?

HW #40page 314 (57–60), page 333 (63, 65, 67, 69, 71, 73, 77, 79)

5.3 continued

Read 319-320

What is the conditional probability formula?

Is it on the formula sheet?

How is it related to the general multiplication rule?

What’s a very common way to lose credit on probability questions?

Movie princesses revisited: In an earlier example, we looked at the relationship between eye color and hair color for movie princesses. What percent of princesses with blonde hair have blue eyes?

More on Pregnancy Tests: Many over-the-counter pregnancy test (OTCPT) kits advertise that they will accurately identify results as early as the first day of a missed period. A positive result on this test indicates that the user is pregnant. However, not all women who test positive actually are pregnant. Suppose that 60% of women who use an OTCPT kit this early are pregnant, the false positive rate is 2%, and the false negative rate is 75%. Make a diagram that will help organize this information.

Space to recopy your diagram, if you want to:

(a) What percent of OTCPT kit users will test positive?

(b) What percent of OTCPT kit users who test positive actually are pregnant?

HW #41:page 333 (81-99 odd, 100, 101)

Chapter 5 Review

FRAPPY: 2011 #2 (Politics and Gender in two towns)

HW #42: page 340 Chapter Review

Chapter 5 Review

HW #43: page 342 Chapter 5 AP Statistics Practice Test

Chapter 5 Test