ap statistics exam review. chapter 6 topics probability law of averages, law of large numbers...
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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
10480 15011 01536 02011 81647 91646 69179 14194 62590 36207 20969 99570 91291 9070022368 46573 25595 85393 30995 89198 27982 53402 93965 34095 52666 19174 39615 9950524130 48360 22527 97265 76393 64809 15179 24830 49340 32081 30680 19655 63348 5862942167 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 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