1. AP Statistics Exam Review
Chapters 6-8

2. Chapter 6 Topics Probability Probability Rules
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

3. 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’

4. 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

5. 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)

6. 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 %
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) = ( ) / 178 = 174 / 178 = 97.8% same as P (Female ∩ No)c = 1 – (4/178)
Gender
Yes No
Total
Male 19 71 90
Female 84 4 88
103 75
178

7. Mutually exclusive (disjoint)
Venn Diagrams
Graphical displays of events, with probabilities
Mutually exclusive (disjoint)
Intersection
Union
A ∩ B

8. 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
103 75
178
Male
Pierced Ears? 19 71 84

9. Conditional Probability
Conditional Probability – the likelihood of an event GIVEN another event is already known (or has occurred)
Examples: Likelihood…
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%
Gender
Yes No
Total
Male 19 71 90
Female 84 4 88
103 75
178

10. Tree Diagrams Gender Yes No Total Male 19 71 90 Female 84 4 88 103 75
178
Gender
Yes No
Total
Male
10.7%
40.0%
50.7%
Female
47.2%
2.1%
49.3%
57.9%
42.1%
100.0%
Gender
Male
Pierced ears
No pierced ears
Female
50.7%
49.3%
95.4%
4.6%
21.1%
78.9%
2.1%
47.2%
40.0%
10.7%

11. 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)
Examples:
P(0,1,2) = P(0)+P(1)+P(2) = = 0.17
P(>4) = P(5)+P(6)+P(7) = = 0.50
P(>1) = P(<1)c = 1 – [ ] = = 0.94
Value 1 2 3 4 5 6 7
Probability
0.01
0.05
0.11
0.15
0.18
0.23
0.17
0.10

12. Discrete Variables Discrete random variables Expected value (mean)
Variance
Standard deviation
Believe it or not, they might ask you to calculate these!  (or, you can use your calculator)
Value 1 2 3 4 5 6 7
Probability
0.01
0.05
0.11
0.15
0.18
0.23
0.17
0.10

13. Continuous Variables Continuous random variables Remember:
Area under the curve, probability of an event
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)

14. 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

15. 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
Subtracting variances
Multiplying variances (by a)

16. 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

17. Binomial Distributions
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

18. Binomial Distributions
Binomial Probability
Interest: the number of successes in ‘n’ independent trials
Formula:
Reads: ‘k’ occurrences out of ‘n’ trials
Example: 50% chance of having a boy… likelihood of having 2 boys out of 5 children

19. Geometric Distributions
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

20. Geometric Distributions
Geometric Probability
Interest: the 1st (or another number) occurrence of a particular event
Formula:
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?

21. 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…