Download Probability: The Foundation of Inferential Statistics

Probability: The Foundation of
Inferential Statistics
October 14, 2009
Subjective Probability
• Throughout the course I have used the word
probability.
– Yet I have not defined it. Instead I have relied on
the assumption that you all have a sense of
probability.
– The book calls this sense of probability “subjective
probability”.
Classical Approach to Probability
• The mathematical Definition of Probability
or
Empirical Approach to Probability
• In classical approach, the parameters are known
– The numbers of cards in a deck. (e.g. the probability of
drawing a king from a full deck of cards)
• In the empirical approach the parameters are not
known.
– Instead we use samples to calculate estimates of
relative probabilities.
Foundations of Empirical Probability
• Discrete vs. Continuous Variables
– Discrete- a variable that is represented in whole numbers.
• Decimals don’t make sense
– Continuous- a variable where intermediate or fractional values
are valid.
• Sample Space
– All possible outcomes that can occur
• In Mendelian genetics, dominant recessive allele Aa located at one
locus may have the following sample space: AA, aa, Aa.
• If Hardy-Weinberg Equilibrium is not violated the relative frequencies
in the population should be:
• f(AA)= 1, p=.25
• f(Aa)= 2, p=.5
• f(aa)=1, p=.25
Mutually vs. Nonmutually Exclusive
Events
• Mutually Exclusive
– Events in our sample that cannot occur together
or overlap.
• Nonmutually exclusive events
– Events in our sample that can occur together.
– A joint probability is the degree in which a set of
events do occur together in a sample.
Calculating Probability
• Probability can be expressed
– in a percentage or relative frequency in 100.
– As a decimal.
• p= .5 is the same as, 50% chance, is the same as saying
50 in one hundred.
• The addition rule: p(A or B)
– It is used when we want to calculate the
probability of selecting an element that has one or
more conditions.
Calculating Probability
• The addition rule: p(A or B) continued
– p(A and B) is the joint probability.
– When the events are mutually exclusive
p(A and B) = 0
The addition rule and odds of dying
• The odds of dying p=1.0
• Usually you die of one
cause.
• What are the odds of
dying of either a plane
accident or a bicycling
accident.
• Bicycling accident= 1 in
4919
• Air/space accident= 1 in
5051
The addition rule and odds of dying
• p(A or B)= .0002 + .0002 – 0= .0004
– -0 because these are mutually exclusive causes of
death.
• .4% chance of dying in a bicycling accident or
a air/space accident.
Multiplication Rule for Independent
Events: p(A) x p(B)
• Used to determine the probability of two or more
events occurring at the same time that are
mutually exclusive.
• Example: You want to know what the odds are
that you will win the lottery. You have to match
all five numbers. The choices range from 1:40.
• The probability for choosing the first number is 1
in 40, the second number 1 in 39, third number 1
in 38…
Joint and Marginal Probabilities
• Joint and marginal probabilities refer to the
proportion of an event as a fraction of the
total.
• To calculate these probabilities we divide the
frequency of the joint or marginal probability
of two or more events by the total frequency.
A
Not A
Marginal Prob.
B
p(A and B)
p(A and B)
p(B)
Not B
p(A and not B)
p(not A and not B)
p(not B)
Marginal Prob.
p(A)
p(not A)
Calculating Probabilities
A
Not A
Marginal Prob.
B
p=f(A and B)/tot
p=f(A and B)/tot
p(B)= f(B)/Total
Not B
p=f(A and not B)/tot
p=f(not A and not B)/tot p(not B)= f(not
B)/Total
Marginal
Prob.
p(A)= f(A)/Total
p(not A)= f(not A)/Total
total/total= 1.00
•The previous graph just gave you where the different types of probabilities are
located on the chart.
•This chart gives you the way you would calculate these probabilites.
•You will be given a frequency for each cell (e.g. B= 20, not B=80, A= 48, not A= 52)
•With this information you should be able to create a similar chart.
Frequency Table Example
A
Not A
B
20
48
not B
80
52
Frequency Table Example
A
Not A
Marginal
Probability
B
20
48
= 20 +48
not B
80
52
= 80 + 52
Marginal
Probability
=20 + 80
=48 +52
Frequency Table Example
A
Not A
Marginal
Probability
B
20
48
68
not B
80
52
132
Marginal
Probability
100
100
Total = 200
Calculating Probabilities
A
Not A
Marginal Prob.
B
p=20/200
p=48/200
p(B)= 68/200
Not B
p=80/200
p=52/200
p(not B)=132/200
Marginal
Prob.
100/200
100/200
200/200= 1.00
Calculating Probabilities
A
Not A
Marginal Prob.
B
p(A and B)=.1
p(not A and B)=.24
p(B)= .34
Not B
p(A and not B)=.4
p(not A and not B)=.26
p(not B)=.66
Marginal
Prob.
p(A)=.5
p(A).5
200/200= 1.00
Conditional Probabilities
A
Not A
B
(not B)
•Conditional probabilities are used when categories are not mutually exclusive.
•The “|” symbol means given.
•Therefore the first cell p(A|B) reads the probability of picking A given B
•example from book A= Alcohol Abuse B= drug abuse. p(A|B) means the probability of
picking an alcohol abuser among drug abusers.
Conditional Probabilities
A
B
(not B)
Not A
p(B|A)≠p(A|B)
A
Not A
Marginal Prob.
B
p(A |B)=.1/.34
p(not A |B)=.24/.34
p(B)= .34
Not B
p(A |not B)=.4/.66 p(not A |not B)=.26/.66
p(not B)=.66
Marginal
Prob.
p(B)=.5
p(A).5
200/200= 1.00
A
Not A
Marginal Prob.
B
p(B |A)=.1/.5
p(B |not A)=.24/.5
p(B)= .34
Not B
p(not B |A)=.4/.5
p(not B |not A)=.26/.5
p(not B)=.66
Marginal Prob.
p(B)=.5
p(A).5
200/200= 1.00
p(B|A)≠p(A|B)
A
Not A
Marginal Prob.
B
p(A |B)=.294
p(not A |B)=.706
p(B)= .34
Not B
p(A |not B)=.606
p(not A |not B)=.394
p(not B)=.66
Marginal
Prob.
p(B)=.5
p(A).5
200/200= 1.00
A
Not A
Marginal Prob.
B
p(B |A)=.2
p(B |not A)=.48
p(B)= .34
Not B
p(not B |A)=.8
p(not B |not A)=.52
p(not B)=.66
Marginal Prob.
p(B)=.5
p(A).5
200/200= 1.00
Determining Joint Probabilites when Conditional
and Marginal Probabilities are given
A
Not A
Marginal Prob.
B
p(B |A)=.2
p(B |not A)=.48
p(B)= .34
Not B
p(not B |A)=.8
p(not B |not A)=.52
p(not B)=.66
Marginal Prob.
p(A)=.5
p(not A).5
200/200= 1.00
A
Not A
Marginal Prob.
B
p(B and A)=.2*.5
p(B and not A)=.48*.5
p(B)= .34
Not B
p(not B and A)=.8*.5 p(not B and not A)=.52*.5
p(not B)=.66
Marginal Prob.
p(A)=.5
200/200= 1.00
p(not A).5
A
Not A
Marginal Prob.
B
p(A and B)=.1
p(not A and B)=.24
p(B)= .34
Not B
p(A and not B)=.4
p(not A and not B)=.26
p(not B)=.66
Marginal Prob.
p(A)=.5
p(A).5
200/200= 1.00
The Binomial Distribution
• Probability of Discrete Sequences
• Lets say you want to know what the probability is
that by chance you can guess 8 out of 10 of a true
false exam.
– For this you would use the formula that describes the
binomial distribution is:
– ! is the symbol for factorial. Example the
4!=4*3*2*1=24
– Don’t worry, I won’t make you calculate these.
Mean and Standard Deviation for a
Binomial Distribution.
• When p=.5 the binomial distribution is
symmetrical and approximates a bell curve.
– This approximation becomes more accurate as N
increases.
• When p<.5 the binomial distribution is
positively skewed.
• When p>.5 it will have a negative skew.
The Binomial Distribution Continued.
• The binomial distribution has all the same
descriptive statistics we already know.
• Mean, standard deviation, and z scores.
• We can use what we already know to relate
this distribution to the normal curve.
Example
• Take the midterms I haven’t passed out yet.
• Let’s say that the mean on this test Is around
.9 and has a standard deviation of .2.
• What is the probability of picking a person at
random who actually failed the test?
Example Continued
•
•
•
•
•
z=(.6-.9)/.2
z=-1.5
Go to the back of the book and see
area beyond z=.0668
Only a 6.6% chance that you failed the test.
Quiz time.