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Probability and Sampling: Part I

What are the odds?What are the odds of finding a newspaper at the news stand?New York timesVillage VoiceJerusalem Post

From a deck of cards:What are the odds of getting the king of spades?What are the odds of getting a king?What are the odds of getting a diamond suit?From ReasonFrom Experience1/52 or 0.01924/52 or 0.076913/52 or 0.2500Sampling and ProbabilityWhat is the probability ofpulling out a red marble ?P( R) = Total number of Red marbles Total number of marblesWhat about probabilities for MULTIPLE events?P( R) = 7/12 or .5833Bag of 7 red and 4 blue, 1 white marbles.Two Rules for Probabilities of Multiple EventsAddition rule: orProbability of a red or a blue marble?Multiplication rule: andProbability of a red and a blue marble?Mutually exclusive eventsNon-mutually exclusive eventsWith replacementWithout replacementAddition Rule: (Part I)What is the probability of getting a white or a red marble?P(W or R) = P(W) + P(R)P(W or R) = 1/12 + 7/12P(W or R) = 8/12 = .6667Bag of 7 red and 4 blue, 1 white marbles.Addition Rule: (Part II)What is the probability of getting a red or a glossy marble?Mutually exclusive events do not require subtractionGlossy marblesP(R or G) = P(R) + P(G)P(R or G) = 7/12 + 2/12 1/12P(R or G) = 8/12 = .6667 P(R & G)Addition Rule: (Part I)What is the probability of getting a white or a red marble?P(W or R) = P(W) + P(R)P(W or R) = 1/12 + 7/12P(W or R) = 8/12 = .6667Can a marble be both white and red at the same time? No, so these are mutually exclusive events, which do NOT require subtraction.Bag of 7 red and 4 blue, 1 white marbles.Multiplication Rule: (Part I)What is the probability of getting a white and then a red marble?If you do put the first marble back.P(W and then R) = P(W) * P(R)P(W and then R) = 1/12 * 7/12P(W and then R) = 7/144 = .0486If you put the objects back after youve taken them out, you have sampled with replacement.Bag of 7 red and 4 blue, 1 white marbles.Bag of 7 red and 4 blue, 1 white marbles.Multiplication Rule: (Part I)What is the probability of getting a white and then a red marble? If you do NOT put the first marble back.P(W and then R) = P(W) * P(R|W)P(W and then R) = 1/12 * 7/11P(W and then R) = 7/132 = .0530If you do not put the objects back after youve taken them out, you have sampled without replacement.Multiplication Rule: (Part II)How to handle sequences of events: What is the probability of reaching into a fresh bag and getting the sequence R, W, B, R, R?P(R) = P(W|R) = P(B|R, W) = P(R|R, W, B) = P(R|R, W, B, R) = 7/121/114/106/95/8Without replacement(7/12)(1/11)(4/10)(6/9)(5/8) = .0088Sum of Two Fair Dice

There is only one way to make a 2two dice in one toss 1 and 1There is only one way to make a 12two dice in one toss 6 and 6Sum of Two Fair Dice

There are 6 ways to make a 7with two dice in one toss:Die 1 = 6, Die 2 =1Die 1 = 5, Die 2 =2Die 1 = 4, Die 2 =3Die 1 = 1, Die 2 =6Die 1 = 2, Die 2 =5Die 1 = 3, Die 2 =4The Probability Distribution of Two Six Sided Dice

Probability.1667.0000.0278.0556.0833.1111.1389The Probability Distribution of Two Six Sided Dice

Probability.1667.0000.0278.0556.0833.1111.1389HTO.5O.5A single event that can go one of two ways -- Two mutually exclusive events.Coin TossExpressed as probability: Two possible outcomes withequal likelihood.

P(H) = = 0.5P(T) = = 0.5Baseline probability

Multiplication rule for calculating the probability of a sequence of outcomes ...H TO.5O.5P(HT) = P(H) * P(T) = (0.5)(0.5) = 0.25

P(TH) = P(T) * P(H) = (0.5)(0.5) = 0.25P(TT) = P(T) * P(T) = (0.5)(0.5) = 0.25H TO.5O.5HTO.5O.5P(HH) = P(H) * P(H)Independent events, Sampling with replacementHHHTTHTTOutcomesProbability of landing on heads twice in a row=Probability of landing on heads on the first flipProbability of landing on heads on the second flipWhat about multiple events? (More than one flip of the coin)P(HH) = 0.25

P(HT) = 0.25P(TH) = 0.25P(TT) = 0.25}P(1 T) = 0.25 + 0.25 = 0.5P(0 T) = 0.25P(2 T) = 0.25# of TailsHHHTTHTTOutcomesAddition rule for calculating the probability of outcomes that are of the same kind:H TO.5O.5H TO.5O.5HTO.5O.5If we plot the outcomes as a histogram we begin to see a familiar shape.

0 1 2 This works for Sequences of any lengthHTO.5O.5

HHHHTTTHHTTT

0 1 2 The Binomial TableA list of ALL the possible outcomes of N events when each event only has two outcomes.Flip a coin 3 times, whats the probability of 0 tails?:

Flip a coin 2 times, whats the probability of 2 tails?:Flip a coin 4 times, whats the probability of 4 tails?:Flip a coin 3 times, whats the probability of 1 head?:Flip a coin 4 times, whats the probability of 3 heads?:.1250.2500.0625.3750.2500Flip a coin 3 times, whats the probability of 2 or more tails?:

Flip a coin 2 times, whats the probability of 1 or less tails?:Flip a coin 4 times, whats the probability of 2 or less tails?:Flip a coin 3 times, whats the probability of 1 or more heads?:.5000.7500.6875.8750The Binomial TableA list of ALL the possible outcomes of N events when each event only has two outcomes.Ever wonder how likely you are to pass a True/False exam if you JUST GUESSED? Assume there are 20 true/false questions on the exam.You need to answer 13 or more correctly to get a 65+..0739 + .0370 + .0148 + .0046 +.0011 + .0002 + .0000 + .0000 = .1316 or 13.16% chance

P(13) + p(14) + p(15) + p(16) + p(17) + p(18) + p(19) + p(20)The Binomial TableA list of ALL the possible outcomes of N events when each event only has two outcomes.What if the baseline probability is not .50?Suppose you are given a coin which you KNOW is weighted: 60% of the time it shows up heads, and 40% of the time it shows up tails.What is the probability of obtaining 3 tails out of 4 flips of this coin?

What is the probability of obtaining 0 heads out of 4 flips of the coin?What is the probability of obtaining 0 tails out of 4 flips of this coin? What is the probability of obtaining 3 heads out of 4 flips of this coin? .1536.1296.3456.0256The Binomial TableA list of ALL the possible outcomes of N events when each event only has two outcomes.What if the baseline probability is not .50?Suppose you are given a coin which you KNOW is weighted: 60% of the time it shows up heads, and 40% of the time it shows up tails.

What is the probability of obtaining 1 or more heads out of 3 flips of the coin?What is the probability of obtaining 3 or fewer heads out of 4 flips of this coin? 3 or fewer heads = 1 or more tails = .87041 or more heads = 2 or fewer tails = .9360The Binomial TableA list of ALL the possible outcomes of N events when each event only has two outcomes.

The Critical Value of an Inferential StatisticCritical Value of the statistic is the value that demarcates the outcomes that will allow us to make conclusions about the data.Chart112345654321

Outcome (sum of two dice)FrequencyPair of Fair Six-Sided Dice

Sheet11234562BinFrequency12345673212345678432345678954345678910654567891011765678910111287698510941110312112121More0

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Outcome (sum of two dice)FrequencyPair of Fair Six-Sided Dice

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Chart112345654321

Outcome (sum of two dice)FrequencyPair of Fair Six-Sided Dice

Sheet11234562BinFrequency12345673212345678432345678954345678910654567891011765678910111287698510941110312112121More0

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Outcome (sum of two dice)FrequencyPair of Fair Six-Sided Dice

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Chart10.250.50.25

Number of TailsProbabilityDistribution for 2 tosses

Sheet1HH0.25HT0.5TT0.25

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OutcomeProbabilityDistribution for 2 tosses

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Chart10.00000095370.00001907350.00018119810.00108718870.00462055210.01478576660.03696441650.0739288330.12013435360.16017913820.1761970520.16017913820.12013435360.0739288330.03696441650.01478576660.00462055210.00108718870.00018119810.00001907350.0000009537

Number TailsPRobabilityBinomial Distribution 20 events

Sheet1NPQ200200.00000095371190.00001907352180.00018119813170.00108718874160.00462055215150.01478576666140.03696441657130.0739288338120.12013435369110.160179138210100.1761970521190.16017913821280.12013435361370.0739288331460.03696441651550.01478576661640.00462055211730.00108718871820.00018119811910.00001907352000.0000009537

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Number HeadsPRobabilityBinomial Distribution 20 events

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Chart10.250.50.25

Number of TailsProbabilityDistribution for 2 tosses

Sheet1HH0.25HT0.5TT0.25

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OutcomeProbabilityDistribution for 2 tosses

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