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Thinking About Probability Outline Basic Idea Different types of probability Definitions and Rules Conditional and Joint probabilities Essentials of understanding stats Discrete and Continuous probability distributions Density Permutations A visit to the Binomial distribution

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outline
Outline
  • Basic Idea
    • Different types of probability
  • Definitions and Rules
  • Conditional and Joint probabilities
    • Essentials of understanding stats
  • Discrete and Continuous probability distributions
    • Density
  • Permutations
  • A visit to the Binomial distribution
  • The Bayesian approach
the problem with probabilities
The Problem with Probabilities
  • Can be very hard to grasp
    • e.g. Monty Hall problem
  • TV show “Let’s make a deal”
  • 3 closed doors, behind 1 is a prize (others have “goats”)
  • Select a door
  • Monty Hall opens one of the remaining doors that does NOT contain a prize
  • Now allowed to keep your original door or switch to the other one
  • Does it make a difference if you switch?
    • http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html
properties of probabilities
Properties of probabilities
  • 0≤ p(A) ≤ 1
    • 0 = never happens
    • 1 = always happens
    • A priori definition
  • p(A) = number of events classifiable as A

total number of classifiable events

    • A posteriori definition
  • p(A) = number of times A occurred

total number of occurrences

properties of probabilities5
Properties of probabilities
  • So:
  • p(A)= nA/N = number of events belonging to subset A out of the total possible (which includes A).
  • If 6 movies are playing at the theater and 5 are crappy but 1 is not so crappy what is the probability that I will be disappointed?
  • 5/6 or p = .8333
probability in perspective
Probability in Perspective
  • Analytic view
    • The common approach: if there are 4 bad movies and one good one I have an 80% chance in selecting a bad one
    • Fisher
  • Relative Frequency view
    • Refers to the long run of events: the probability is the limit of chance i.e. in a hypothetical infinite number of movie weekends I will select a bad movie about 80% of the time
    • Neyman-Pearson
  • Subjective view
    • Probability is akin to a statement of belief and subjective e.g. I always seem to pick a good one
    • Bayesian
some definitions
Some definitions
  • Mutually exclusive1
    • both events cannot occur simultaneously
    • A + !A = impossible
  • Exhaustive sets
    • set includes all possible events
    • the sum of probabilities of all the events in the set = 1
some definitions8
Some definitions
  • Equal likelihood: roll a fair die each time the likelihood of 1-6 is the same; whichever one we get, we could have just as easily have gotten another
    • Counter example- put the numbers 1-7 in a hat. What’s the probability of even vs. odd?
  • Independent events:
    • occurrence of one event has no effect on the probability of occurrence of the other
laws of probability addition
Laws of probability: Addition
  • The question of Or
  • p(A or B) = p(A) + p(B)
    • Probability of getting a grape or lemon skittle in a bag of 60 pieces where there are 15 strawberry, 13 grape, 12 orange, 8 lemon, 12 lime?
    • p(G) = 13/60 p(L) = 8/60
    • 13/60 + 8/60 = 21/60 = .35 or a 35% chance we’ll get one of those two flavors when we open the bag and pick one out
laws of probability multiplication
Laws of probability: Multiplication
  • The question of And
  • If A & B are independent
  • p(A and B) = p(A)p(B)
  • p(A and B and C) = p(A)p(B)p(C)
    • Probability of getting a grape and a lemon (after putting the grape back) after two draws from the bag
    • p(Grape)*p(Lemon) = 13/60*8/60 = ~.0288
conditional probabilities and joint events
Conditional Probabilities and Joint Events
  • Conditional probability
    • One where you are looking for the probability of some event with some sort of information in hand
    • e.g. the odds of having a boy given that you had a girl already.1
  • Joint probability
    • Probability of the co-occurrence of events
    • E.g. Would be the probability that you have a boy and a girl for children i.e. a combination of events
  • In this case the conditional would be higher because if we knew there was already a girl that means they’re of child-rearing age, able to have kids, possibly interested in having more etc.
conditional probabilities
Conditional probabilities
  • If events are not independent then:
  • p(X|Y) = probability that X happens given that Y happens
    • The probability of X “conditional on” Y
  • p(A and B) = p(A)*p(B|A)
  • Stress and sleep relationship conditioned on gender
    • Little relation for fems, negative relation for guys
  • The observed p-value at the heart of hypothesis testing is a conditional probability
    • p(Data|H0)
joint probability
Joint probability
  • When dealing with independent events, we can just use the multiplicative law.
  • Joint probabilities are of particular interest in classification problems and understanding multivariate relationships
    • E.g. Bivariate and multivariate normal distributions

?

simpson s paradox
Simpson’s paradox
  • Success rates of a particular therapy
  • What’s wrong with this picture?
  • Is the treatment a success?
discrete probability distribution
Discrete probability distribution
  • Involves the distribution for a variable that takes on only a few values
  • Common example would be the Likert scale
continuous probability distribution
Continuous probability distribution
  • We often deal with continuous probability distributions in inference, the most famous of which is the normal distribution
  • The height of the curve is known as the density
  • We expect values near the ‘hump’ to be more common
permutations
Permutations
  • Counting is a key part of understanding probability (e.g. we can’t tell how often something occurs if we don’t know how many events occur in general).
  • Some complexity arises when we consider whether we track the order and whether events are able to be placed back for future selection.1
  • How many ways can a set of N units be ordered?
    • Factorial
  • Permutations of size k taken from N objects
    • Ordered, without replacement
    • There are 5 songs on your top list, you want to hear any combination of two. How many pairs of songs can you create? In this case ab != ba, i.e. each ordering counts
      • 20
permutations18
Permutations
  • Combinations: finding the number of combinations of k objects you can choose from a set of n objects
    • Unordered, without replacement
  • In this case, any pair considered will not be considered again
    • i.e. ab = ba
  • From our previous example, there are now only 10 unique pairs to be considered
  • The combination described above will come back into play as we discuss the binomial
the binomial
The Binomial
  • Bernoulli trials = 2 mutually exclusive outcomes
  • Distribution of outcomes
  • Order of items does not matter
  • Only the probability of various outcomes in terms of e.g. numbers of heads and tails
  • N = # trials = 3
coin toss
Coin toss
  • How many possible outcomes of the 3 coin tosses are there?
  • List them out: HHH HHT HTT TTT TTH THH THT HTH
  • Now condense them ignoring order
    • e.g. HTT = THT = flips result in only 1 heads
  • What is the probability of 0 heads, 1 heads, 2 heads, 3 heads?
distribution of outcomes22
Distribution of outcomes
  • Now how about 10 coin flips?
  • That’d be a lot of work writing out all the possibilities.
  • What’s another way to find the probability of coin flips?
  • Use the formula for combinations
binomial distribution
Binomial distribution
  • Find a probability for an event using:
  • N = number of trials
  • r = number of ‘successes’
  • p = probability of ‘success’ on any trial
  • q = 1-p (probability of ‘failure’)
  • CNr=The number of combinations of N things taken r at a time
slide24

So if I want to know the odds of getting 9 heads out of 10 coin flips or p(H,H, H,H, H,H, H,H, H,T):

  • p(9) =
  • 10(.001953)(.5)=.0098 = .01
using these probabilities
Using these probabilities
  • What is the probability of getting 4 or fewer heads in 10 coin tosses?
  • Addition
    • p(4 or1 less) = p(4) + p(3) + p(2) + p(1) + p(0) =
    • .205 + .117 + .044 + .010 + 001 =
    • p = .377
    • About 38% chance of getting 4 or fewer heads on 10 flips
test a hypothesis
Test a Hypothesis
  • Now take it out a step.
  • Suppose you were giving some sort of treatment to depressed individuals and assumed the treatment could work or not work, and in general would have a 50/50 chance of doing so if it wasn’t anything special (i.e. just a placebo). Then it worked an average of 9 times out 10 administrations.
  • Would you think there was something special going on or that it was just a chance occurrence based on what was expected?
  • p = p(9) + p(10) = .011
not just 50 50
Not just 50/50
  • Not every 2 outcome situation has equal probabilities associated with each option
  • There are two parameters we are concerned with when considering a binomial distribution
    • 1. p = the probability of a success. (q is 1-p)
    • 2. n = the number of (Bernoulli) trials
  • More info about binomial distribution
    • m = Np
    • s2=Nqp
      • In R
        • Rcmdr (Distribution menu)
        • ?pbinom (command line)
  • Approximately “normal” curve when:
  • p is close to 0.5
    • If not then “skewed” distribution
  • N large
    • If not then not as representative a distribution
examples
Examples
  • Small N p = .8 N = 10
bayesian probability
Bayesian Probability
  • Thomas Bayes (c. 1702 –1761)
  • The Bayesian approach involves weighing the probability of an event by prior experience/knowledge, and as such fits in well with accumulation of knowledge that is science.
  • As new evidence presents itself, we will revise our previous assessment of the likelihood of some event
  • Prior probability
    • Initial assessment
  • Posterior probability
    • Revised estimate
bayesian probability31
Bayesian Probability

With regard to hypothesis testing:

p(H0) = probability of the null hypothesis

p(D|H0) = the observed p-value we’re used to seeing, i.e. the probability of the data given the null hypothesis

p(H1) = probability of an alternative1

p(D|H1) = probability of the data given the alternative hypothesis

empirical bayes method in statistics
Empirical Bayes method in statistics
  • Bayesian statistics is becoming more common in a variety of disciplines
  • Advantages: all the probabilities regarding hypothesis testing make sense, interval estimates etc. are what we think they are and what they are not in null hypothesis testing
  • Disadvantage: if the priors are not well thought out, could lead to erroneous conclusions
  • Why don’t we see more of it?
    • You actually have to think of not only ‘non-nil’ hypotheses but perhaps several viable competing hypotheses, and this entails:
      • Actually knowing prior research very well1
      • Not being lazy with regard to the ‘null’, which now becomes any other hypothesis
  • We will return with examples regarding proportions and means later in the semester.
summary
Summary
  • While it seems second nature to assess probabilities, it’s actually not an easy process in the scientific realm
  • Knowing exactly what our probability regards and what it does not is the basis for inferring from a sample to the population
  • Not knowing what the probability entails results in much of the misinformed approach you see in statistics in the behavioral sciences