AP Statistics Exam Review
Chapter 6 Topics

Probability
 Law of averages, Law of Large Numbers
 Randomness, Simulations

Probability Rules
 Probability Model – Sample space, event
 Two events – mutually exclusive (disjoint),
union/intersection, Addition rule
 Two-way tables, Venn Diagrams
 Conditional probability
 Tree diagrams, Multiplication rule
Probability

Probability – the likelihood (chance) of a
particular outcome to occur (0-1 or 0-100%)
 Short-run (independent, random)
vs. Long-run (law of large numbers)
 Myth of ‘law of averages’
Probability

Simulation – imitation of chance behavior,
based on a model that accurately reflects the
situation
 State (the question of interest)
-> Plan (determine chance device)
-> Do (repetitions)
-> Conclude
 e.g., Use of Table of Random Digits
10480 15011 01536 02011 81647 91646 69179 14194 62590 36207 20969 99570 91291 90700
22368 46573 25595 85393 30995 89198 27982 53402 93965 34095 52666 19174 39615 99505
24130 48360 22527 97265 76393 64809 15179 24830 49340 32081 30680 19655 63348 58629
42167 93093 06243 61680 07856 16376 39440 53537 71341 57004 00849 74917 97758 16379
Probability Rules

Probability Model
Event – any collection of outcomes from some chance process
Sample space (S) – all possible outcomes
Probability of each outcome, P (A)
Mutually exclusive (disjoint) events P(A or B) = P(A) + P(B)
Example: Rolling 2 dice
Sample space, S {1-1, 1-2…, … 5-6, 6-6}
36 possible outcomes, each equally likely
 Comments:





○ Make sure and identify the ENTIRE sample space
Can shorten by use of “…” if a pattern is formed
○ Remember: all possible outcomes = 100%
○ Note: It may be easier to calculate the Complement of an event
P(Ac)
Probability Rules

Two-way Tables, Probability
 Example: How common is pierced ears among college
students?
 Comments:
○ Calculate totals (if needed)
○ May need to convert to %
Gender
Yes
No
Total
Male
19
71
90
Female
84
4
88
103
75
178
Total
 Calculations:
 Mutually exclusive (Addition rule)
e.g., P(Male + Yes) = P(Male ∩ Yes) = 19/178 = 10.7%
 Not-mutually exclusive (Addition rule)
e.g., P(Male or Yes) = P(Male ∪ Yes)
= P(Male) + P(Yes) – P(Male ∩ Yes)
= (90+103-19) / 178 = 174 / 178 = 97.8%
same as P (Female ∩ No)c = 1 – (4/178)
Venn Diagrams

Graphical displays of events, with probabilities
Mutually exclusive
(disjoint)
Intersection
A∩B
Union
Venn Diagrams

Continuing with our example…
 Note: You cannot diagram
ALL of the combinations
 Diagram the topic of
interest (Males / ears)
Gender
Yes
No
Total
Male
19
71
90
Female
84
4
88
Total
103
75
178
Male
Pierced
Ears?
19
71
84
Conditional Probability
Conditional Probability – the likelihood of an
event GIVEN another event is already known
(or has occurred)
Gender
Yes
No
Total
Male
19
71
90
 Examples:
Female
84
4
88
Likelihood…
Total
103
75
178

 Has pierced ears given that he is male
P(Yes I Male) = 19 / 90 = 21.1%
 Is female given that she has pierced ears
P(Female I Yes) = 84 / 103 = 81.6%
Tree Diagrams
Gender
Yes
No
Total
Gender
Yes
No
Total
Male
19
71
90
Male
10.7%
40.0%
50.7%
Female
84
4
88
Female
47.2%
2.1%
49.3%
Total
103
75
178
Total
57.9%
42.1% 100.0%
21.1%
10.7%
50.7%
78.9%
95.4%
40.0%
47.2%
49.3%
4.6%
2.1%
Variables and Distributions

Random Variables
 Discrete – fixed set of possible values
 Continuous – all values within an interval
(e.g., set of all numbers between 0 and 1)

Probability Distribution
 All possible values + their probabilities (extension of
Probability Model: sample space, event)
Value

Probability
0
1
2
3
4
5
6
7
0.01
0.05
0.11
0.15
0.18
0.23
0.17
0.10
 Examples:
○ P(0,1,2) = P(0)+P(1)+P(2) = 0.01+0.05+0.11 = 0.17
○ P(>4) = P(5)+P(6)+P(7) = 0.23+0.17+0.10 = 0.50
○ P(>1) = P(<1)c = 1 – [0.01+0.05] = 1-0.06 = 0.94
Discrete Variables

Discrete random variables
 Expected value (mean)
 X   xp  x1 p1  x2 p2  ...  xn pn  4.31
 Variance
 X2   ( x   X )2 p  (1  4.31)2 *0.05  (2  4.31)2 *0.11  ...( xn  X )2 pn  0.0052
 Standard deviation  X   X2  0.072
4.31  0.072
Believe it or not, they might ask you to calculate these! 
(or, you can use your calculator)
Value
Probability
0
1
2
3
4
5
6
7
0.01
0.05
0.11
0.15
0.18
0.23
0.17
0.10
Continuous Variables

Continuous random variables
Z
 Area under the curve, probability of an event

X 

Remember:
 X values are ALONG the curve
 Z-values are ‘normalized’ standard deviation values
 The +/- 1, 2, 3 values are standard deviations away
from the mean
 Percentiles correspond to
the area under the curve
(50% is mean; 84% is +1
standard deviation mean)
Continuous Variables

Z-values…
 Z-scores are ‘mirrored’ +/- from the mean values
 Z-scores give values to the left of the line
○ P(Z > 1.4), find the z-value, then subtract from 1
Z
X 

Transforming Random Variables

Effects of changing the values for the mean and
standard deviation
 Mean 
○ Adding, subtracting, multiplying values to the mean do
EXACTLY that… add/subtract/multiply to the mean
○ Also, does not change the shape of the distribution
 Variance 
○ Adding variances
2
○ Subtracting variances
   2   1 2
2
1
2
2
 12   2 2   1 2 2
○ Multiplying variances (by a)
a * 1
2
2
Transforming Random Variables

Effects of changing the values for the mean and
standard deviation

 Standard deviation
○ For adding, subtracting, multiplying standard deviations
 They operate in the same manner as variances but…
 You MUST work with variances 1st…
then take the square-root of the values!
 Also, changing the variance (and standard deviation) values by
adding/subtracting/multiplying DOES change the shape
Binomial Distributions

Binomial
 Perform several independent trials of the same chance
process and record number of times a particular outcome
occurs
 Examples:
○ Toss a coin 5 times
○ Spin a roulette wheel 10 times
○ Random sample of 100 babies born in U.S.
 Sample problems:
○ Shuffle deck, turn over top card, repeat 10 times, count how
many aces are observed
○ Choose students at random and identify how many are taller
than 6 feet
○ Flip a coin, repeat 20 times, count how many heads are
observed
Binomial Distributions

Binomial Probability
 Interest: the number of successes in ‘n’ independent
trials
 Formula:
n
k 
 
p (1  p)
k
nk
n
n!
where   
 k  k !(n  k )!
○ Reads: ‘k’ occurrences out of ‘n’ trials
○ Example: 50% chance of having a boy… likelihood of
having 2 boys out of 5 children
5
2
 2  0.50
 
(1  0.50)
5 2
5  4  3  2 1

 (0.50)2  (0.50)3  10  (0.50)5  0.3125
2 1  3  2 1
Geometric Distributions

Geometric
 Perform independent trials of the same chance process
and record the number of trials until a particular outcome
occurs
 Examples:
○ Roll a pair of dice until you get doubles
○ Attempt a 3-point shot until one is made
○ Placing a $1 bet on number 15 on roulette until you win
 Sample problems:
○ Shuffle deck, turn over top card, repeat until the 1st ace is
observed
○ Firing at a target, record the number of shots needed to hit a
bulls-eye
○ Generally an 80% free throw shooter, record the number of
shots needed to miss his 2nd shot
Geometric Distributions

Geometric Probability
 Interest: the 1st (or another number) occurrence of a
particular event
 Formula:
P(Y  k )  (1  p)k 1  p
○ Reads: ‘k’ occurrence, based on ‘p’ probability
○ Example: Japanese ‘sushi roulette’ has ¾ of the slots
showing sushi, ¼ slots showing wasabi. What is the
likelihood a person spinning the wheel hits the 1st
wasabi slot on the 3rd spin?
P(Y  3)  (1  0.25)31  0.25  0.1406
Binomial and Geometric Distributions

Suggestions…
 Which to use? Look for ‘clue words’ –
○ How many occurrences… Binomial
○ 1st time / 3rd occurrence… Geometric
 Write down the formula(s)
○ The formula will often tell you what you need…
e.g., probability, occurrences/trials
 Read the problems carefully…
○ Frequently they will ask:
i.e., what is the probability of at least 3 occurrences?
what is the probability of seeing an ace no earlier than the
4th card?
○ In both situations, calculate multiple probability values and
then add them…
P(Y  3)  P(Y  4)  P(Y  5)  P(Y  6)...
or...1  [ P(1  Y  3)]