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## PowerPoint Slideshow about 'Thinking About Probability' - bernad

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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

- 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

- 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 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

- 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

- 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 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

- 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

- 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 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

- 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

- 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

- Success rates of a particular therapy
- What’s wrong with this picture?
- Is the treatment a success?

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

- 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

- 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

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

- 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

- 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 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

- 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

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

- 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

- 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 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

- Small N p = .8 N = 10

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 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

- 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

- 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

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