Review of Descartes rational method
 

All Horses are Mammal<-----premise 1
All Mammals are Animals<-----premise 2
____________________________________

 All Horses are Animals <-----conclusion
 
 
 
 
 
  Some Connections

• The scientific method
• Further observing and experimenting
• Refining and retesting explanations
• Probabilities
• Here's an analogy:
• Rules:deduction
• Probability:induction
• Probability will be important for the rest of the course, especially when
• we get into the stats.
• Probability
• What is Probability?
• Reasoning with probabilities
• The law of large numbers

• There is a 90% chance of rain.
• I'm 75% sure I'm right about this.
• There is a 50% chance that my parents are coming to visit this weekend
• 10% of children born with disease develop lung cancer.
• The probability that this coin will come up heads is 50%
• What do these statements mean?
• Types of Probabilities
• Some probabilities are proportions of elements drawn from a set
• Probability of side effect given disease
• Long-run probability of having this side effect
• Percentage of actual cases having this side effect
• Probability of rain
• Percentage of forest area expected to be covered by rain.

More types of probabilities
 
 

• Single event probabilities
• Some events occur only once
• Cannot be viewed as coming from a set
• Must construct some kind of set
• There is a 35% chance I failed this test.
• Under circumstances like these, I have failed 35% of the tests I have taken in the past.

Communicative Probabilities

• Often we use the language of probability
• 50% certain means "I don't know"
• 0% certain means "No way"
• 100% certain means "Definitely"
• We don't use fine-grained probabilities
• I am 32.4% certain that this event will occur

How is probability used?

• Probability plays an important role in statistics
• How likely is that a given event was due to chance?
• That is a question we will try to answer
• Probability distributions will be important
• Discrete probability distributions
• Events can only have certain values (e.g., Heads/Tails)
• Continuous probability distributions
• Events can take on any possible values (e.g., means)
• Random Variable
• A variable whose value is a numerical outcome of a random phenomenon
• The variable X, where X is the number of heads resulting from four coin flips

What does this mean?

• What is the relative likelihood of getting a head on a coin flip?
• What if we tried 4 coin flips 96 times?
• How many times should you get 4 heads?
• E(N)=Probability*Trials
• E(N)=0.0625*96=6

    This is a binomial distribution...

Distributions

• Discrete vs. Continuous
• Distribution contains the space of all outcomes
• Probability density function
• Cumulative density function
• We'll talk a lot more about distributions later

How do probabilities combine?

• Disjoint events
• Both A and B cannot occur at the same time?
• Probability of A or B
• Pr(A or B)=Pr(A) + Pr(B)
• Probability I will go to Emo's A=.5
• Probability I will go to Hole in the Wall B=.27
• P(A or B)=.5+.27=.77
• What is the probability I will go to another club?

Independent events

• Two events are independent if the outcome of one event does not determine the outcome of another.
• Pr(A | B)=Pr(A)
• iid: identical independently distributed
• Two successive coin flips are independent
• Two spins of a roulette wheel are independent
• The election of the President and the members of Congress are not independent
• They share common causal factors

Probability of two independent events

• How likely is it that two independent events will occur?
• Pr(A & B)=P(A)*P(B)
• Probability of getting two heads on two successive flips
• Pr(Head)=.5
• Pr(2 heads)=.5*.5=.25
• The probability of a conjunction is always less than or equal to the probability of either conjunct.

The conjunction fallacy

Linda is 31 years old, single, and bright.She majored in philosophy.As a
student, she was deeply concerned with issues of social justice and
participated in demonstrations.

How likely is it:

• Linda is a banker?
• Linda is a banker and active in the feminist movement?

The mean of a random variable

• The mean of the random variable is the average of possible values of the variable.
• For discrete random variables sum of (Values*Probabilities)
• (.0625*0) + (.25*1) + (.375*2) + (.25*3) + (.0625*4)=2
• For continuous random variables
• Need calculus to calculate areas
• Mean is the point where the density curve would balance
• Looking at Multiple events
• Flipping a coin 4 times
• If I did this 3 times, what would happen?
• Three heads the first time
• 0 the second time
• 1 the third time
•   Even though 2 heads is the most likely event, I might not witness it in 3 tries.

The law of large numbers

• If I made 1000 sets of 4 coin flips
• The distribution ought to start to look more like the one I showed before
• The distribution of a random variable will look right in the long  run
• That's why we run lots of subjects!
• There is no law of small numbers
• If I see 8 heads in a row, that does not increase the probability that I will see a tail on the next flip
• Gambler's fallacy

Probabilities will be important

• We will use probability to determine how likely it is that an observation was due to chance
• We will rely on various probability distributions
• The Normal distribution (z)
• Student's t distribution
• The F distribution