Transcript Probability and Binomial distribution

Remember empirical (measured) frequency distributions
http://www.nwfsc.noaa.gov/publications/survey/2001/2001fig1.html
Probability distribution: theoretical frequency
distribution, the distribution that you expect to see
based on theory or other knowledge
If your data do not conform to the exception it can
suggest biological mechanism
Basic Probability
Read very carefully S&R 5.1 or Zar Ch 5
S&R is better on this topic
Important concepts
How to calculate probabilities
Difficulty in obtaining random and impendent samples
Probability deals with experiments that can be performed
over and over again with individual outcomes that are
unpredictable, but whose long term average outcomes can
be determined
When these long term averages are turned into
percentages they are called probabilities
The set of all possible outcomes is called the sample space
and any subset of the sample space is called an event.
http://www.scit.wlv.ac.uk/university/scit/maths/calculus/modules/topics/precalc/probabil/learn.htm
Example: trapping students
-4 kinds of students AU (70%), AG (26%), FU (1%), FG
(3%)
-Proportions are known from enrollment record.
-Rarely know.
-What proportions would you get if you “trapped” at
different locations on campus?
So, how does one take a random (all members of
population equally likely to be sampled) sample of
people, fish, trees, etc…………
If we could randomly sample students we would
expect ~ 3% of them to be FG.
Probability ranges from 0-1
P[FG]=0.03
The 4 categories of students encompass the set
“Trapping” one or a group of students is an event
Trapping 3 students could yield the event
{AU, AG, FU}
Trapping any American student includes AU and AG, call it
A
Trapping a graduate student includes AG and FG, call it B
The intersection of A and B written A  B includes only
the events that are shared by A and B……. So only AG
The union of A and B includes events which are either A
or B or both, written A  B……. So {AG, AU, FG}
What is the probability of trapping an AG? 0.26
What is the probability of trapping a student who is
either American or a graduate student?
don’t
P[A  B]= P[A]+ P[B]- P[A  B]
include
P[A  B]=P{AU,AG}+P{AG,FG)-P{AG} twice
P[A  B]= (0.70+0.26)+(0.26+0.03)-0.26
P[A  B]= 0.99
Could have figured this out by knowing that P[FU] was 0.01!
Up till know, talking about “sampling” one student.
If we want to catch 2, we must catch one first and then
a second.
Must decide whether or not to replace the first one
In very large populations, it doesn’t really matter.
Are the two students caught independent? Doesn’t
relate to whether mom & dad still pay tuition!
Does the probability of catching student 1, affect the
probability of catching student 2?
Events are independent if:
P[DB]= P[D] P[E]
P [intersection of D and E] = P[D] P[E]
P [events shared by D and E] = P[D] P[E]
(in English) the P of one event has no effect on the P of
the second event
So……if events are independent, the probability of
each events are multiplied to determine the joint
probability
The probability of catching 2 FG’s is
P [FG]* P[FG]
0.03*0.03=.0009 (very small)
In the above case, it doesn’t matter which FG you
catch first (for our purposes) one FG is as good as
another
A dresser drawer contains one pair of socks of each of the
following colors: blue, brown, red, white and black. Each pair
is folded together in matching pairs. You reach into the sock
drawer and choose a pair of socks without looking. The first
pair you pull out is red -the wrong color. You replace this pair
and choose another pair. What is the probability that you will
choose the red pair of socks twice?
P [red red]= (0.2*0.2)=0.04
But, P [red] is always 0.2 (if you put the socks back)
Every time you draw, P=0.2. Only when you talk about a
specific string are the Ps multiplied
Hint: don’t put the red sox back in if you don’t want to choose them again.
Alternatively open you eyes while getting dressed.
Patient: Will I survive this risky operation?
Surgeon: Yes, I'm absolutely sure that you will survive
the operation.
Patient: How can you be so sure?
Surgeon: Well, 9 out of 10 patients die in this operation,
and yesterday my ninth patient died.
If deaths are independent, the patient still has a 90% chance of dying!
Distinguishable events
What is the P of catching an AU and an FG in your
sample?
Is it P[AU]*P[FG]?
0.7*0.03?
NO
You could trap (AU FG) or (FG AU), there are 2 ways to
do this, so P of catching an AU and an FG is
2 P[AU]*P[FG]
2 * 0.7 * 0.03=0.0420
A small tangent……..
When does order matter? Combinations vs.
Permutations
Combination: order does not matter
Permutation: order does matter
Combinations locks are misnamed. They are really
permutation locks.
Permutation: An ordered selection of k objects from a set of
n objects is called a permutation of n objects, k at a time
The following formula can be used to count permutations.
n!
(n-k)!
5! = five factorial = 5*4*3*2*1=120
http://www.scit.wlv.ac.uk/university/scit/maths/calculus/modules/topics/precalc/probabil/learn.htm
Combination: A selection of k objects from a set of n
objects is called a combination of n objects, k at a time
The following formula can be used to count combinations.
n!
(n-k)! k!
Another small tangent……..
Conditional probability: The occurrence of one event
is somehow dependent on the occurrence of another
event.
In terms of our student trapping study, what is the
probability of being a graduate student, given that
the student is American?
P[A  B] P [intersection of A and B]
P
A,
given
B
P[AlB]=
=
=
P[B]
P[B]
P[AG]
P[AlB]=
=
P[american]
0.26
0.96
A generalized version of conditional probability is
Bayes’ theorem
Very trendy in statistics
More on topic later
The binomial distribution is used when there are exactly
two mutually exclusive outcomes of a trial. These
outcomes are appropriately labeled "success" and
"failure".
The binomial distribution is used to obtain the
probability of observing x successes in N trials, with the
probability of success on a single trial denoted by p. The
binomial distribution assumes that p is fixed for all
trials.
Such a success/failure experiment is also called a
Bernoulli experiment or Bernoulli trial.
Must simplify previous sampling to consider only whether
students are foreign or American (ignore grad vs UG)
binomial distribution = only 2 options
In the probability space {p, q} p=[F] and q=P[A]
So, compute the probability space of samples of 2 students
{FF, FA, AA}
{p2, 2pq, q2}
The probability of catching 2
FG’s is
P [FG]* P[FG]
0.03*0.03=.0009 (very small)
P of catching an AU and an
FG is
2 P[AU]*P[FG]
2 * 0.7 * 0.03=0.0420
Look at pattern:
Sample 2
{FF, FA, AA}
{p2,
2pq,
q2}
Sample 3
{FFF, FFA, FAA, AAA}
{p3, 3p2 q, 3pq2, q3}
.The formula for expanding the binomial function is:
where
Ex pg 73 S & R
Insect population with 40% infection. Sample 5 insects.
What distribution of samples is expected ? P infection
independent for each insect.
p = P [infected] = 0.04
q = P [not infected] =0.06
{p5, 5p4 q, 10 p3q2, 10 p2q3 , 5pq4, q5}
Probability of
5 infected
insects
Probability of
5 “clean”
insects
Insect population with 40% infection. Sample 5 insects.
What distribution of samples is expected ? P infection
independent for each insect.
p = P [infected] = 0.04
q = P [not infected] =0.06
{p5, 5p4 q, 10 p3q2, 10 p2q3 , 5pq4, q5}
(0.4)5 + 5(0.4)4 (0.6) + 10(0.4)3 (0.6)2 + 10 (0.4)2 (0.6)3 + 5(0.4) (0.6)4 + (0.6)5
Probability of
5 infected
insects
Probability of
5 “clean”
insects
Combine probabilities with number of samples taken
If you took 5 insects from the population x times….
Predict the number of times you would get 5
infected……….5 clean
900
800
expected
frequency
700
600
500
400
300
200
100
0
zero
one
two
three
number infected insects
four
five
Compare your expected frequency to a real,
observed frequency when you actually collect the
samples
How well do they match up?
Ex pg 74 S & R
900
800
expected
frequency
700
observed
600
500
400
300
200
100
0
zero
one
two
three
number infected insects
four
five
There are tests for determining whether observed
frequencies differ from expected by more than can be
attributed to chance alone (maybe covered later).
For now, think about it visually.
Usually, we don’t know p and q
If you have a theoretical basis for p and q (eg. 50:50 sex
ratio) you can compare observed to predicted