introduction to probability - harvard...

Introduction to probability

P(A) = Probability of an event, A, occuring

Before we start discussion distributions, let’s take a stepback and talk about some basic rules of probability.Probability is fundamentally about assigning probabilitiesto events.An event can be pretty much anything for which there is analternative outcome.Eg) A = { sun rises tomorrow, Supreme Court nomineewill be blocked, etc. }

Professor Jason Anastasopoulos [email protected] University of Georgia

[POLS 4150] Intro. to Probability Theory, Discrete and Continuous Distributions

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Rules of probability

P(AC) = 1− P(A)

P(AC) Probability of something not happening.P(A) Probability of something happening.



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Complements of events

If P(Terrorism) = 0.01.What is P(Terrorismc) =?



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Union of events

P(A or B) = P(A) + P(B)

A ⊥ B

Probability that one event or another event that areindependent from each other is just their sum.



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Union of events

If P(Terrorism) = 0.01 and;P(Falcons win superbowl) = 0.05What is:P(Terrorism or Falcons win superbowl) =?



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Union of events

If P(Terrorism) = 0.01 and;P(Falcons win superbowl) = 0.05

P(Terrorism or Falcons win superbowl) =

P(Terrorism) + P(Falcons win superbowl) =

0.01 + 0.05 = 0.06



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Intersection of events

P(A and B) = P(A)P(B)

A ⊥ B

Probability of both independent events occurring is justtheir probabilities multiplied together.



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If P(Terrorism) = 0.01 and;P(Falcons win superbowl) = 0.05What is:P(Terrorism and Falcons win superbowl) =?



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If P(Terrorism) = 0.01 and;P(Falcons win superbowl) = 0.05

P(Terrorism and Falcons win superbowl) =

P(Terrorism)P(Falcons win superbowl) =

(0.01)(0.05) = 0.0005



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Probability distributions for discrete and continuousvariables

Probability distributions are full distributions of all possibleoutcomes and probability of those outcomes occurring.Recall that discrete variables and variables which take ona finite number of values.Continuous variables take on a theoretically infinitenumber of values.



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Probability distributions for discrete and continuousvariables

Discrete probability distributions are probabilitydistributions which assign a probability to each individualoutcome.Continuous probability distributions are probabilitydistributions which assign probabilities to intervals.



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Probability distribution of a discrete variable

Children in families = y ={4,6,2,1,1,2}

The number of children in families is a good example of adiscrete variable.



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Probability distribution of a discrete variable

Children in families = y ={4,6,2,1,1,2}0 ≤ P(y) ≤ 1

N∑i=1

P(y) = 1

P(y = 4) = 1/6, P(y = 6) = 1/6,P(y = 2) = 2/6,P(y = 1) = 2/6This is the full probability distribution of y .



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Probability distribution for continuous variables

Continuous variables have a theoretically infinitecontinuum of values.Strangely enough, because of this continuous distributionsalways assign probabilities to ranges rather than values.



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Probability distribution for continuous variables: IQscores

Let’s take IQ scores again as an example.Each IQ range corresponds to a probability value.



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P(100 > IQ > 160) = 0.50P(100 < IQ < 40) = 0.50



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These probability values actually directly correspond to the68− 95− 99 rule if the data follow a normal distribution.



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x̄ = 100, s = 15P(85 > IQ > 115) = 0.68



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x̄ = 100, s = 15P(70 > IQ > 140) =?



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x̄ = 100, s = 15P(70 > IQ > 140) = 0.95



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x̄ = 100, s = 15P(55 > IQ > 155) =?



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Summarizing probability distributions

At their core, probability distributions are just functions justlike those you might remember from calculus: ie f (x) = x2

They are functions that are defined, however, by theirparameters.These parameters are typically the mean and variance



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Normal distribution= f (µ, σ) = N(µ, σ)

Normal distribution= f (x̄ , s) = N(x̄ , s)

For example, where the normal distribution lies on thex–axis depends upon it’s mean, or expected value.How fat the normal distribution is depends on its standarddeviation (or variance which is just s2



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VClinton,GAcounties ∼ N(36,5)

When we describe a variable in terms of its distribution, weusually specify what kind of distribution it follows, the meanand the standard deviation of that distribution.VClinton,GAcounties variable is the county vote share for HillaryClinton in the 2016 election for the 159 counties in GA.



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Clinton vote share in GA counties



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Trump vote share in GA counties



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Trump vote share in GA countiesDef: If VTrump,GA is a variable containing Trump vote share for

GA counties.Def: And VTrump,GA ∼ N(60,5)

Q: What are a and b in the equationP(a < VTrump,GA < b) = 0.95? What does this mean, inwords?



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Q: What are a and b in the equationP(a < VTrump,GA < b) = 0.95? What does this mean, inwords?

A: a = 60− 2 ∗ 5 = 50, b = 60 + 2 ∗ 5 = 70.



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Q: What is p in the equation P(55 < VTrump,GA < 65) = p?What does this mean, in words?



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Q: What is p in the equation P(55 < VTrump,GA < 65) = p?What does this mean, in words?

A: p = 0.68.



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introduction to probability - harvard...

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