CHAPTERS 6-8AP Statistics Exam Review

Chapter 6 Topics Probability

Law of averages, Law of Large NumbersRandomness, Simulations

Probability RulesProbability Model – Sample space, eventTwo events – mutually exclusive (disjoint),

union/intersection, Addition ruleTwo-way tables, Venn DiagramsConditional probabilityTree 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 situationState (the question of interest)

-> Plan (determine chance device)-> Do (repetitions)

-> Concludee.g., Use of Table of Random Digits

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Probability Rules Probability Model

Event – any collection of outcomes from some chance processSample space (S) – all possible outcomesProbability 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 %

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)

Gender Yes No Total

Male 19 71 90

Female 84 4 88

Total 103 75 178

Venn Diagrams Graphical displays of events, with probabilities

Mutually exclusive(disjoint)

Intersection Union

A ∩ B

Venn Diagrams Continuing with our example…

Note: You cannot diagram ALL of the combinations

Diagram the topic ofinterest (Males / ears)

Gender Yes No Total

Male 19 71 90

Female 84 4 88

Total 103 75 178

Male Pierced Ears?

7119

84

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 maleP(Yes I Male) = 19 / 90 = 21.1%

Is female given that she has pierced earsP(Female I Yes) = 84 / 103 = 81.6%

Gender Yes No Total

Male 19 71 90

Female 84 4 88

Total 103 75 178

Tree DiagramsGender Yes No Total

Male 19 71 90

Female 84 4 88

Total 103 75 178

Gender Yes No Total

Male 10.7% 40.0% 50.7%

Female 47.2% 2.1% 49.3%

Total 57.9% 42.1% 100.0%

Gender

Male

Pierced ears

No pierced ears

Female

Pierced ears

No pierced ears

50.7%

49.3%

95.4%

4.6%

21.1%

78.9%

2.1%

47.2%

40.0%

10.7%

Variables and Distributions Random Variables

Discrete – fixed set of possible valuesContinuous – 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) = 0.01+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

Value 0 1 2 3 4 5 6 7

Probability 0.01 0.05 0.11 0.15 0.18 0.23 0.17 0.10

Discrete Variables Discrete random variables

Expected value (mean)

Variance

Standard deviation

1 1 2 2 ... 4.31X n nxp x p x p x p

2 0.072

4.31 0.072X X

2 2 2 2 2( ) (1 4.31) *0.05 (2 4.31) *0.11 ...( ) 0.0052X X n X nx p x p

Value 0 1 2 3 4 5 6 7

Probability 0.01 0.05 0.11 0.15 0.18 0.23 0.17 0.10

Believe it or not, they might ask you to calculate these! (or, you can use your calculator)

Continuous Variables Continuous random variables

Area under the curve, probability of an event Remember:

X values are ALONG the curveZ-values are ‘normalized’ standard deviation valuesThe +/- 1, 2, 3 values are standard deviations away

from the meanPercentiles correspond to

the area under the curve (50% is mean; 84% is +1standard deviation mean)

XZ

Continuous Variables Z-values…

Z-scores are ‘mirrored’ +/- from the mean valuesZ-scores give values to the left of the line

○ P(Z > 1.4), find the z-value, then subtract from 1

XZ

Transforming Random Variables Effects of changing the values for the mean and

standard deviationMean

○ Adding, subtracting, multiplying values to the mean do EXACTLY that… add/subtract/multiply to the mean

○ Also, does not change the shape of the distributionVariance

○ Adding variances

○ Subtracting variances

○ Multiplying variances (by a)

2

2 2 21 2 1 2

2 2 21 2 1 2

2 21*a

Transforming Random Variables Effects of changing the values for the mean and

standard deviationStandard deviation

○ For adding, subtracting, multiplying standard deviationsThey 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:

○ Reads: ‘k’ occurrences out of ‘n’ trials○ Example: 50% chance of having a boy… likelihood of

having 2 boys out of 5 children

(1 )

!

!( )!

k n knk p p

n nwhere

k k n k

2 5 2 2 3 55 0.502

5 4 3 2 1(1 0.50) (0.50) (0.50) 10 (0.50) 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:

○ 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?

1( ) (1 )kP Y k p p

3 1( 3) (1 0.25) 0.25 0.1406P Y

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/trialsRead 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… ( 3) ( 4) ( 5) ( 6)...

...1 [ (1 3)]

P Y P Y P Y P Y

or P Y