\5.1 Randomness and Probability
Complete the following activities with a partner.
Activity 1 The “1 in 6 wins” Game
Activity2: “Whose Book is This”?
Suppose that 4 friends get together to study at Tim’s house for their next test in AP Statistics. When
they go for a snack in the kitchen, Tim’s three-year-old brother makes a tower using their textbooks.
Unfortunately, none of the students wrote his name in the book, so when they leave each student takes
one of the books at random. When the students returned the books at the end of the year and the clerk
scanned their barcodes, the students were surprised that none of the four had their own book. How
likely is it that none of the four students ended up with the correct book?
1. On four equally sized slips of paper, write book 1, book 2, book 3, book 4 (or use names!)
2. Shuffle the papers and lay them down one at a time in a row. If the book number on the
paper matches its position in the row (e.g. book 2 ends up in the second position), this
represents a student choosing his own book from the tower of textbooks. Count the number
of students who get the correct book.
3. Repeat this several more times, recording the number of students who get the correct book in
each trial.
4. Combine your results with your classmates and estimate how often none of the four end up
with their own book. Keep track of proportion of 0’s as we go.
Read 283-286
What is the law of large numbers?
If we observe more and more repetitions of any chance process, the proportion of timies that a specific
outcome occurs approaches a single value.
How do you interpret a probability?
Long run relative frequency…or predicted relative frequency in a large number of trial.
Always between 0 and 1, 0 means impossible, 1 means guaranteed
Beware of scientific notation on your calculators!
The probability of getting a sum of 7 when rolling two dice is 1/6. Interpret this value.
If you roll two die many times the probability of getting a sum of 7 will occur 1 in 6 times.
Read 287-289
What are some myths about randomness?
Random phenomena is not predictable in the short run.
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Don’t fall for “laws of averages” (You might have 8 girls one day!!)
Don’t confuse the law of large numbers, which describes the big idea of probability, with the “law of
averages”.
Read 289, 291–2
What is the purpose of simulation?
To estimate probabilities when they are difficult to calculate theoretically.
What are the four steps? Do you need to do the four steps for each simulation?
Four Steps for the “Whose Book Is This?
State: When four students mix up their AP stats books, what is the probability that when each student
randomly chooses a book, he doesn’t end up with his own?
Plan: On four equally sized slips of paper, write the numbers 1, 2, 3, 4,. Shuffle the papers and lay them
down one at a time in a row. If the number on the paper matches its position in the row, this represents a
student choosing is own book from the mixed-up books.
Do: Do this process many times, recording the number of students who get the correct book in each
trial.
Conclude: Out of 30 total repetitions, there were ____times when none of the students ended up with
their own book. So the estimated probability is __/30 = ____%
Alternate Example: Streakiness
Suppose that a basketball announcer suggests that a certain player is streaky. That is, the announcer
believes that if the player makes a shot, then he is more likely to make his next shot. As evidence, he
points to a recent game where the player took 30 shots and had a streak of 7 made shots in a row. Is this
evidence of streakiness or could it have occurred simply by chance? Assuming this player makes 48%
of his shots and the results of a shot don’t depend on previous shots, how likely is it for the player to
have a streak of 7 or more made shots in a row?
HW #1 page 293 (1, 3, 9, 11, 25*, 27*) *Do at least 10 trials
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5.1 Simulations
Read 290-291
Complete CYU page 286 and page 292
HW #2 page 295 (15, 17, 19, 23, 29*)
*Do at least 10 trials
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5.2 Probability Rules
Read 299-301
What is a sample space?
The sample space of a chance process is the set of all possible outcomes.
Roll two dice – sample space (page 300) 36 outcomes
What is a probability model?
A description of some chance process that consists of a sample space S and a probability for each
outcome is a probability model.
What is an event?
An event is any collection of outcomes from some chance process.
P(sum is 5), rolling two dice = 4/36 or 1/9
Imagine flipping a fair coin three times. Describe the probability model for this chance process and
use it to find the probability of getting at least 1 head in three flips.
Read 301-303
What does it mean if two events are mutually exclusive?
Two events are ME (or disjoint) if they have no outcomes in common.
 ME
P( 5) and P(snake-eyes),
 Not ME
P(5) and P(difference of 1),
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Summarize the five basic probability rules. Page 302
Alternate Example: AP Statistics Scores
Randomly select a student who took the 2010 AP Statistics exam and record the student’s score.
Here is the probability model:
Score
1
2
3
4
5
Probability 0.233 0.183 0.235 0.224 0.125
(a) Show that this is a legitimate probability model.
(b) Find the probability that the chosen student scored 3 or better.
(c) Find the probability that the chosen student didn’t get a 1.
HW #3: page 297 (30*, 31-36), page 309 (43, 45, 47)
*do at least 10 trials
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5.2 Two-Way Tables
Read 303-305
What is the general addition rule? Is it on the formula sheet? What if the events are mutually exclusive?
P(A or B) = P(A) + P(B) – P(A and B)
*Formula sheet on exam is different, they use union which implies (or)
Alternate Example: Who Owns a Home?
What is the relationship between educational achievement and home ownership? A random sample of
500 people who participated in the 2000 census was chosen. Each member of the sample was identified
as a high school graduate (or not) and as a home owner (or not). Overall, 340 were homeowners, 310
were high school graduates, and 221 were both homeowners and high school graduates.
(a) Create a two-way table that displays the data.
Suppose we choose a member of the sample at random. Find the probability that the member
(b) is a high school graduate.
(c) is a high school graduate and owns a home.
(d) is a high school graduate or owns a home.
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Read 305-308
Alternate Example: Phone Usage
According to the National Center for Health Statistics, in December 2008, 78% of US households had a
traditional landline telephone, 80% of households had cell phones, and 60% had both. Suppose we
randomly selected a household in December 2008.
(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.
(c) Find the probability that the household has at least one of the two types of phones.
P(at least one) =.98
(d) Find the probability that the household has neither type of phone.
P(neither)=1-((.78+.8)-.6) = .02
(e) Find the probability the household has a cell phone only.
P(cell phone only) = .20
”.
HW #4 page 309 (49–55 odd)
HW #5 page 276 Cumulative AP Practice Test (1–17)
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5.3 Conditional Probability and Independence
Read 312-314
What is a conditional probability?
The probability one event happens given that another event is already known to have happened.
P (B|A)
Conditional probability is not commutative
P (B|A) ≠P(A|B)
Example Page 312
Event A- is male
Event B-has pierced ears
P(A) = 90/178
P(A and B) = 19/178
P(B) = 103/178
P(A or B) = 174/178
Conditional Probability
P(A|B) = P (male given has pierced ears) = 19/103 = 18.4%
P(B|A) = P (has pierced ears given is male) = 19/90 = 21.1 %
Read 315-316
How can you tell if two events are independent?
Two events A and B are independent if the occurrence of one event has no effect on the chance that the
other event will happen.
In other words, event A and B are independent if P(A|B) =P(A) and P(B/A) = P(B)
Example of Independent –
Toss a fair coin twice, event A – first toss is head, event B – second toss is head, P(B/A) = P(B)
Example of not independent –
A male, B pierced ears, P(B|A) ≠ P(B) 19/90 ≠ 103/178
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Read 317-320
What is the general multiplication rule? Is it on the formula sheet?
P(A and B) = P(A) times P(B|A) , no, it is not on the formula sheet
Rule of thumb – (or , add) (and, multiply)
Alternate Example: Media Usage and Good Grades
In January 2010, the Kaiser Family Foundation released a study about the influence of media in the lives
of young people ages 8-18. In the study, 17% of the youth were classified as light media users, 62%
were classified as moderate media users and 21% were classified as heavy media users. Of the light
users who responded, 74% described their grades as good (A’s and B’s), while only 68% of the
moderate users and 52% of the heavy users described their grades as good. According to this study,
what percent of young people ages 8-18 described their grades as good?
State: What percent of young people say they get good grades?
Plan:
Do: P(good grades) = (.17)(.74) + (.62)(.68) + (.21)(.52) = .6566
Conclude: about 66% of the students in the study described their grades as good.
Read 321-323
What is the multiplication rule for independent events? How is it related to the general multiplication
rule?
P(A and B) = P(A) times P(B)
If the events are independent then P(B/A) = P(B)
What’s the difference between “mutually exclusive” and “independent”?
Two events are mutually exclusive if they cannot occur at the same time. These events cannot be
independent
Independent – the occurrence of one event has no effect on the chance that the other event will happen.
HW #6 page 311 (57-60), page 329 (63, 65, 67, 69, 73, 77, 79)
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5.3 continued
Read 324-327
What is the conditional probability formula? Is it on the formula sheet? How is it related to the general
multiplication rule?
General Probability Rule
P(A and B) = P(A) ∙ P(B|A)
Not on AP Exam Formula Sheet!
Conditional Probability Formula
P(B/A) = P(A and B)
P(A)
On the Exam Formula Sheet!!
What’s a very common way to lose credit on probability questions?
Show Work! You must at least show the fraction not just the decimal.
Alternate Example: Media Usage and Good Grades
In an earlier alternate example, we looked at the relationship between media usage and grades for youth
ages 8-18. What percent of students with good grades are heavy users of media?
State: What percent of yout with good grades are heavy users of media?
Plan:
P(heavy users/have good grades) = P(heavy users and have good grades)
P(good grades)
Do: (use tree diagram) = .1092/.6566 =.166
Conclude: or about 17%
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Alternate Example: False Positives and Drug Testing
Many employers require prospective employees to take a drug test. A positive result on this test
indicates that the prospective employee uses illegal drugs. However, not all people who test positive
actually use drugs. Suppose that 4% of prospective employees use drugs, the false positive rate is 5%,
and the false negative rate is 10%.
(a) What percent of prospective employees will test positive? .036 + .048
(b) What percent of prospective employees who test positive actually use illegal drugs?
Tree Diagram
P(took drugs/positive test) = P(took drugs and tested positive) =
P(tested positive)
.036
.036 + .048
=
.429
About 43% of the prospective employees who test positive took drugs
HW #7 page 330 (83*, 85, 87, 91*, 93, 95, 97, 99, 104-106) *don’t need 4-steps for section 5.3
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Learning Objective
Interpret probability as a long-run relative
frequency.
Use simulation to model chance behavior.
Determine a probability model for a chance
process.
Use basic probability rules, including the
complement rule and the addition rule for
mutually exclusive events.
Use a two-way table or Venn diagram to
model a chance process and calculate
probabilities involving two events.
Use the general addition rule to calculate
probabilities.
Calculate and interpret conditional
probabilities.
Use the general multiplication rule to
calculate probabilities.
Use tree diagrams to model a chance
process and calculate probabilities
involving two or more events.
Determine whether two events are
independent.
When appropriate, use the multiplication
rule for independent events to compute
probabilities.
Section
Related
Example
on Page(s)
Relevant
Chapter
Review
Exercise(s)
5.1
284
1
5.1
290, 291
2
5.2
299
4, 11
5.2
302
5, 11
5.2
303, 306, 307
5, 6, 8, 9
5.2
307
5, 6
5.3
313, 324, 325
7, 9
5.3
319, 320
7
5.3
320, 325
7
5.3
316
8, 9
5.3
321, 322, 323
10, 11
Can I do
this? 1 = not
at all
10 = you bet
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