Chapter 6 - Probability
Assignments:
a.
b.
c.
d.
e.
f.
g.
3, 6, 10, 12
15, 19, 23, 26, 27
28, 33, 36, 38
41, 42, 43, 47
50, 51, 53, 55
64, 68, 73
75 and Activity 6.3: The “Hot Hand” in Basketball
Objectives:
Know how to find probability as it relates to area.
Know the difference between classical and experimental probability.
Know the probability formulas from the formula sheet and what they are for.
Know how to give the practical interpretation of a stated probability. What will happen in the long run?
This is also called the relative frequency interpretation.
Know how to do the probability of “at least one”.
Know how to assign digits to create a simulation of a probability experiment. This involves the concepts
of replacement and non-replacement, independence and dependence.
Know the relationships between mutually exclusive (disjoint), not mutually exclusive (not disjoint),
independent, and dependent. Are disjoint events with probabilities greater than zero independent or
dependent?
Know how to compute simple “and” and “or” probabilities.
Define disjoint events.
Set up tables to calculate probabilities.
Set up trees to calculate probabilities.
Find probability using relative frequency.
What are the “magic words” when dealing with relative frequency?
Use tree diagrams and tables in conjunction to find probabilities, including conditional probabilities.
Basic Terms
Chance Experiment: _______________________________________________________
Sample Space: ___________________________________________________________
Event: __________________________________________________________________
Simple Event: ____________________________________________________________
Union: _________________________________________________________________
Intersection: _____________________________________________________________
Complementary Events: ____________________________________________________
Disjoint (Mutually Exclusive) Events: _________________________________________
Venn Diagrams
1. Draw a Tree Diagram for all the possible outcomes of first tossing a coin and then rolling a 6sided die.
2. Do assignment a: 3, 6, 10, 12
Approaches to probability
Classical
PE  
number of outcomes favorable to E
number of outcomes in the sample space
1. What is the probability of tossing tails and then rolling a 5?
2. If I randomly draw a ball from a box containing 3 orange balls and 5 green balls, what is the
probability that I will draw an orange ball? What is the probability that I will not draw an orange
ball?
3. If I randomly throw a dart at a target with three scoring areas consisting of concentric circles with
radii of 2cm, 4cm and 6cm, what is the probability of the dart landing in the outer ring?
Relative Frequency
5
The Law of Large Numbers: ________________________
1. Toss a coin 100 times and graph the relative frequency of heads after each toss.
Subjective
A personal measure of strength of belief.
1. What do you believe is the probability of Oklahoma beating Oklahoma State in football this year?
2. What do you believe are your chances of making a 3 or better on the AP Statistics exam?
Basic properties of probability:
Remember all probabilities are between 0 and 1.
If S is the sample space (every outcome possible) then P(S) = 1.
Complementary events have probabilities that add up to 1. P(-E) = 1 – P(E)
Do assignment b: 15, 19, 23, 26, 27
Addition Rule (disjoint events)
If two events E and F are disjoint, then P(EF) = (E or F) = P(E) + P(F).
1. What is the probability of rolling a 3 on a pair of fair dice? What is the probability of rolling a 7?
What is the probability of rolling a three or a 7?
2. If I draw a card from a well shuffled deck. What is the probability of drawing a 2 or a face card?
3. Are drawing a king and drawing a red card from a deck of well shuffled cards disjoint events?
4. If you draw a card and roll 2 6-sided dice, are drawing an ace and rolling an 8 disjoint events?
Addition to the Addition Rule
If two events E and F are not disjoint, then…
P(E or F) = P(E) + P(F) – P(E and F)
1. What is the probability of rolling an even number or a number greater than 3 on a 6-sided die?
2. What is the probability of drawing either a spade or a face card?
Conditional Probability
What is the probability that an extra on Star Trek dies given that he is wearing a red shirt?
– This is an example of a conditional probability.
The probability of event E occurring given that event F has already occurred can be found
either by direct reasoning or by the formula:
PE | F 
P(E and F)
P(F)
1. What is the probability of drawing a King of Spades from a well shuffled deck?
2. What is the probability of drawing a King given that you’ve drawn a spade from a well shuffled
deck? (Or…If you draw a spade what is the probability that it is a king?)
3. Given the following table of values from a survey of teachers on job satisfaction…
 Find the probability associated with each cell.

Find the probability that a teacher is satisfied with his job given that he is a high school
teacher.

Find the probability that a teacher is a high school teacher given that he is satisfied with his
job.
College
High School
Elementary
Total
Do assignment c: 28, 33, 36, 38
Satisfied
74
224
126
Unsatisfied
43
171
140
Total
117
395
266
Independence
Given events E and F, E and F are independent if P(EF) = P(E). That is to say the
probability of E occurring remains the same whether F has occurred or not.
Are the events independent?
E = Mr. Moore is out of the room during a televised OU game.
F = OU makes a good play.
Since P(Mr. Moore is out of the room during a televised OU game) does not change whether OU makes a
good play or not, E and F are independent.
Note that if P(EF) = P(E) then P(FE) = P(F). Independence works both ways.
Also, if E and F are independent and F happens or doesn’t happen, either way, it doesn’t effect the
probability of E happening or not happening for that matter. If E and F are independent nothing we
know about F effects anything involving the likelihood of E and vise-versa. So…
P(E-F) = P(E), P(-EF) = P(-E), and P(F-E) = P(F), P(-FE) = P(-F)
Multiplication Rule (independent events)
Given independent events E and F, the P(EF) = P(E and F) = P(E)P(F)
1. The probability that Mr. Moore will be out of the room during a televised OU game is .16. The
probability that OU will make a great play is .37. Find the probability that OU will make a great
play and Mr. Moore will be out of the room during a televised OU game.
2. If you roll a 6-sided die 5 times, what is the probability that you roll the same number every time?
3. What is the probability of drawing an even numbered card from a well shuffled deck and rolling a
10 on a pair of dice?
4. If I randomly pick a student from our class and have them roll a pair of dice, what is the
probability that a guy will roll “snake-eyes!”?
5. Are disjoint events independent?
6. If I draw 5 cards in row without replacing the cards each time, are the draws independent? If I
replace the cards and reshuffle each time are they independent?
7. P(rolling a 4 on a die and drawing a face card and tossing heads) =
Do assignment d: 41, 42, 43, 47
General Probability Rules
Addition Rule
For events E and F,
P(E or F) = P(E) + P(F) – P(E and F)
Example: 60% of all students eat lunch in the main lunchroom. 40% eat lunch in the Commons. 25% of
all students eat lunch both places. Find the probability that a randomly selected student eats lunch either
in the lunchroom or the Commons.
P(lunchroom) = .6
P(Commons) = .4
P(lunchroom and Commons) = .25
So…
P(lunchroom or Commons) =
P(lunchroom) + P(Commons) – P(lunchroom and Commons) = .6 + .4 - .25 = .75
1. What is the probability that a student doesn’t each lunch either place?
2. What is the probability that a student eats in only one of the two places?
Multiplication Rule
For events E and F,
P(E and F) = P(EF)P(F)
Example: 30% of those attending prom last year did so without a date. Of those without dates, only 13%
actually asked someone to go with them as a date. If a student attending prom last year were randomly
selected, what is the probability that he or she didn’t have a date and asked someone to prom? (These are
not independent. Why?)
P(dateless) = .3
P(askeddateless) = .13
P(asked) = unknown
We can do either
P(dateless and asked) = P(datelessasked)P(asked)
or
P(asked and dateless) = P(askeddateless)P(dateless)
Since we don’t know all of the info for the first method we’ll use the second.
P(asked and dateless) = .13.3 = .039
Thus 3.9% of the students at prom asked someone to prom and were dateless on prom night.
Using a Tree Diagram to calculate probabilities.
When you use a tree diagram you can multiply along the branches to calculate the end probabilities. For
example…
P(heads and 6) =
P(tails and 4 or heads and 2) =
Rolling a die
1
heads
2
3
4
5
6
Tossing a coin
1
2
3
tails
4
5
6
65% of the senior class at MHS has contracted senioritis. A new test has been devised to diagnose this
debilitating disease. The test isn’t perfect, however, returning false positives 2% of the time and false
negatives 1.5% of the time.
1. Draw the tree diagram for this scenario.
2. If I select a senior at random find…
a. P(correct diagnosis)
b. P(false diagnosis)
3. Twenty percent of all passengers who fly from Los Angeles to New York do so on Airline G.
This airline misplaces luggage for 10% of its passengers, and 90% of this lost luggage is
subsequently recovered. If a passenger who has flown from LA to NY is randomly selected, what
is the probability that the selected individual flew on Airline G, had luggage misplaced, and
subsequently recovered the misplaced luggage?
Do assignment e: 50, 51, 53, 55
Conditional Probability, Tables & Tree Diagrams
1) A survey of MHS students found that 36% said that they would be interested in going
Saturn. Of those who wanted to go to Saturn, 60% were not seniors. Of those who
not want to go to Saturn, 30% were seniors.
Create a tree diagram for this situation.
What is the probability that a randomly selected
(a) Student wanted to go to Saturn?
(b) Was a senior and wanted to go to Saturn?
(c) Student was a senior?
(d) Senior wanted to go to Saturn?
(e) Saturn wannabe was a senior?
Fill in the two-way (contingency) percent table with the information.
Saturn
Not Saturn
Senior
Not senior
Totals
Totals
100%
If 500 students were surveyed, fill in the two-way (contingency) counts table with the information.
Saturn
Senior
Not senior
Totals
Not Saturn
Totals
500
to
did
2) When the male students at MHS were asked, 50% said they do not date someone from MHS. When the
female students were asked, 40% said they do not date someone from MHS. The male students make up
52% of the student population.
Draw a tree diagram to represent this situation.
Fill in the two-way (contingency) percent table with the information.
Date MHS
student
Don’t date MHS
student
Male
Female
Totals
Totals
100%
What is the probability that a randomly selected
(a) Student does not date someone from MHS?
(b) Student is female?
(c) Student is female and does not date someone from MHS?
(d) Student who dates someone from MHS is male?
If 500 students were surveyed, fill in the two-way (contingency) counts table with the information.
Date MHS
student
Male
Female
Totals
Don’t date MHS
student
Totals
500
(e) How many were male?
(f) How many were females who did not date someone from MHS?
3) Mr. Moore’s Statistics classes collected information on their
gender and handedness. The two-way table below gives the
percents of the statistics students who fall into each category.
Right-handed
Left-handed
Totals
Male
.35
.15
.5
Female
.3
.2
.5
Totals
.65
.35
100%
If a student is chosen at random from the Statistics classes, what is the probability of a
(a) Student being left-handed, if the student is a male?
(b) Student being female, if the student is right-handed?
(c) Student being female and right-handed?
If there are 140 Statistics students, (I’m dreaming now) complete the two-way counts table below.
Male
Female
Totals
Right-handed
Left-handed
Totals
(d) How many are male and left-handed?
Draw a tree diagram with correct probabilities to represent this situation? There are two possible tree
diagrams depending on which event you chose for your initial branches. Challenge: Work both!
4) The following counts were provided concerning people in rural, suburban, and
urban areas and the length of their vacations (1-7 days or 8+ days).
1-7 days
8+ days
Totals
Rural
80
43
123
Suburban
39
32
71
Urban
37
19
56
Totals
156
94
250
Complete the two-way percent table with the information.
Rural
1-7 days
8+ days
Totals
Suburban
Urban
Totals
100%
What is the probability that a randomly selected
(a) Person spends 8+ days on vacation, given that he/she is from a rural area?
(b) Person is from a rural area, given that he/she spends 8+ days on vacation?
(c) Person spends 1-7 days on vacation, given that he/she is from a suburban area?
(d) Person is from an urban area, given that he/she spends 1-7 days on vacation?
Can you draw a tree diagram with correct probabilities to represent this
situation?
5) We wish to look at probabilities concerning whether students own a dog or have a cell
phone. We poll 400 students at MHS. One hundred thirty-two own a dog. Eightyfour students owning a dog have a cell phone. One hundred forty students
without a dog have a cell phone.
Complete the two-way table of counts for this situation.
Cell Phone
Yes
No
Have Dog
No Dog
Totals
Totals
400
Complete the two-way percent table for this situation.
Cell Phone
Yes
No
Totals
Have Dog
No Dog
Totals
100%
Create a tree diagram for this situation using whether or not student owns a dog is the first branch.
(a) What is the probability that a student has a dog and a cell phone?
(b) What is the probability that a student has a dog, if he/she has a cell phone?
(c) What is the probability that a student has a cell phone or a dog?
(d) How many students with dogs have cell phones?
(e) How many students with cell phones have dogs?
(f) How many students have a cell phone and a dog?
Do assignment f: 64, 68, 73
SIMULATION

OH GIRL! A couple plans on having children until they have a girl OR until they have a max of four
children, whichever happens first.
SIMULATE
1. Assign the digits, 0 through 9, to represent birth of child being a boy or girl in this situation.
2. Describe how you will run the simulation using a random number generator given above.
4. Run your simulation 15 times, recording the results. Complete the table below.
X=
Tally
P(X =
)
1 or G
2 or BG
3 or BBG
4 or BBBG
4 or BBBB
5. What is the probability that there will be a girl in amongst the children, based on your simulation?
6. What is the probability that the couple has exactly two children, based on your simulation?
7. What is the probability that the couple has at least two children, based on your simulation?
8. What is the probability that the couple had fewer than two children, based on your simulation?
WORKING WOMEN. Assume that the percentage of women in the labor force of a large metropolitan
area is 40%. A company hires ten workers, two of whom are women. Is this likely?
SIMULATE
1. Assign the digits, 0 through 9, to represent the “what’ in this situation.
2. Describe how you will run the simulation using a random number generator.
3. Run your simulation 20 times, recording the results in the table below.
X=
0
1
2
3
4
5
6
7
Tally
P(X = )
8
9
10
4. What is the probability that this would occur by chance, based on your simulation?
5. Based on your simulation, estimate the probability that this company would employ two or fewer
women of ten selected, by chance?
6. Estimate the expected number of women of ten new workers this company would employ, making
use of your simulation results.
8. Based on your simulation, is there a potential legal battle brewing? Justify.
STATISTICAL EVIDENCE OF DISCRIMINATION: In 1974, 48 male bank supervisors were each
given the same personnel file and asked to judge whether the person should be promoted or the file held
for other applicants interviewed (not promoted at this time). The files were identical except that half of
them were classified as a female worker and the other half were classified as male workers. Of the 24
“male worker” files, 21 were recommended for promotion. Of the 24 “female worker” files, 14 were
recommended for promotion. [B. Rosen and T. Jerdee, “Influence of Sex Role Stereotypes on Personnel
Descisions,” Journal of Applied Psychology, 59, 1974, pages 9-14]
Is this convincing evidence that the bank supervisors discriminate against female applicants? Or could the
difference in the numbers recommended for promotion reasonably be attributed to chance alone? That is,
there was no discrimination…it just happened that 21 out of the 35 bank supervisors who recommended
promotion got files marked “male worker.”
SIMULATE
1. Define the random variable.
2. Assign the digits, 0 through 9, to represent “what” in this situation described.
3. Describe how you would carry out your simulation using 35 numbers generated by a random
number generator.
4. Carry out your simulation 20 times.
5. If the simulation was run 100 times, the
following histogram displays the findings. Do
you believe your simulation provides evidence
that the bank supervisors discriminated against
females? Use the results to argue your case.
Do assignment f: 75 and Activity 6.3: The “Hot Hand” in Basketball