Introduction to Probability

1
Introduction to Probability

• Experiments, Counting Rules, and Assigning
  Probabilities
• Events and Their Probability
• Some Basic Relationships of Probability
• Conditional Probability
• Bayes Theorem

2
Probability

• Probability is a numerical measure of the
  likelihood that an event will occur.
• Probability values are always assigned on a scale
  from 0 to 1.

3
An Experiment and Its Sample Space

• An experiment is any process that generates well
  defined outcomes.
• The sample space for an experiment is the set of
  all experimental outcomes.
• A sample point is an element of the sample space,
  any one particular experimental outcome.

4
A Counting Rule for Multiple-Step Experiments

• Experiment - a sequence of k steps in which
  there are
• first step - n1 possible results,
• second step - n2 possible results,
• and so on,
• The total number of experimental outcomes is
  given by (n1)(n2) . . . (nk).
• Graphical representation of a multiple step
  experiment is a tree diagram.

5
Example

• A linear arrangement of three deoxyribonucleic
  aced (DNA) nucleotides is called a triplet. A
  nucleotide may contain any one of four possible
  bases adenine (A), cytosine (C), guanine (G),
  and thymine (T). How many different triplets are
  possible?
• If a diploid cell contains three pairs of
  chromosomes, and one member of each pair is found
  in each gamete, how many different gametes are
  possible?

6
Number of experimental outcomes when x objects
are to be selected from a set of N
objects.Number of combinations of N objects
taken x at a timewhere N! N(N - 1)(N - 2) .
. . (2)(1) x! x(x - 1)( x - 2) . . .
(2)(1) 0! 1Of a total of 25 samples, 6 are
to be used in an experiment. How many different
combinations of 6 samples can be chosen?
Counting Rule for Combinations
7
Counting Rule for Permutations

• Number of experimental outcomes when x objects
  are to be selected from a set of N objects where
  the order of selection is important.
• Number of permutations of N objects taken x at a
  time
• In how many ways can 12 different amino acids be
  arranged into a polypeptide chain of 5 amino
  acids?

8
Assigning Probabilities

• Classical Method
• Assigning probabilities based on the assumption
  of equally likely outcomes.
• Relative Frequency Method
• Assigning probabilities based on experimentation
  or historical data.
• Subjective Method
• Assigning probabilities based on the assignors
  judgment.

9
Classical Method

• If an experiment has n possible outcomes, this
  method would assign a probability of 1/n to each
  outcome.
• Example
• Experiment Rolling a die
• Sample Space S 1, 2, 3, 4, 5, 6
• Probabilities Each sample point has a 1/6
  chance of occurring.

10
Relative Frequency Method

• Experiment or survey is repeated under exactly
  the same conditions n times.
• Event A is observed to occur k times then
• P(A)
• Example
• The four human blood types are genetic
  phenotypes. Of 5400 individuals examined, the
  following frequency of each blood type is
  observed. What is the relative frequency of each
  blood type.
• Blood type Frequency
• O 2672
• A 2041
• B 486
• AB 201

11
Subjective Method

• It might be inappropriate to assign probabilities
  based solely on historical data.
• Uses any data available as well as experience and
  intuition, but ultimately a probability value
  should express the degree of belief that the
  experimental outcome will occur.
• The best probability estimates often are obtained
  by combining the estimates from the classical or
  relative frequency approach with the subjective
  estimates.

12
Some Basic Relationships of Probability

• There are some basic probability relationships
  that can be used to compute the probability of an
  event without knowledge of all the sample point
  probabilities.
• Complement of an Event
• Union of Two Events
• Intersection of Two Events
• Mutually Exclusive Events

13

• Complement of an Event
• The complement of event A is defined to be the
  event consisting of all sample points that are
  not in A.
• The complement of A is denoted by Ac.
• Union of Two Events
• The union of events A and B is the event
  containing all sample points that are in A or B
  or both.
• The union is denoted by
• A ??B

14

• Intersection of Two Events
• The intersection of events A and B is the set of
  all sample points that are in both A and B.
• The intersection is denoted by
• A ???
• Addition Law
• The addition law provides a way to compute the
  probability of event A, or B, or both A and B
  occurring.
• The law is written as
• P(A ??B) P(A) P(B) - P(A ? B?

15

• Mutually Exclusive Events
• Two events are said to be mutually exclusive if
  the events have no sample points in common. That
  is, two events are mutually exclusive if, when
  one event occurs, the other cannot occur.
• Addition Law for Mutually Exclusive Events
• P(A ??B) P(A) P(B)

16

• Conditional Probability
• The probability of an event given that another
  event has occurred is called a conditional
  probability.
• The conditional probability of
• A given B is denoted by
• P(AB)
• A conditional probability is computed as follows

17

• Multiplication Law
• The multiplication law provides a way to compute
  the probability of an intersection of two events.
• The law is written as
• P(A ? ?B) P(B)P(AB)

18

• Independent Events
• Events A and B are independent if
• P(AB) P(A).
• Multiplication Law for Independent Events
  
• P(A ? B) P(A)P(B)
• The multiplication law also can be used as a test
  to see if two events are independent.

19

• Example
• Either allele A or a may occur at a particular
  genetic locus. An offspring receives one of its
  alleles from each of its parents. If one parent
  possesses alleles A and a and the other parent
  possesses alleles a and a, what is the
  probability of an offspring receiving
• (a) an A and an a
• (b) two a alleles
• (c) two A alleles

20
Example

• Consider the following biological sequence
• ATAGTAGATACGCACCGAGGA
• If we wish to assess the probability of such a
  sequences, we might let
• P(A) pA P(C)pC P(G)pG P(T)pT
• And assume independence so that
• P(ATAGTAGATACGCACCGAGGA)
• pApT pA pG pG pA
• pA8pC4pG6pT3
• However the assumption of independence may not be
  correct

21
Example

• Consider the following data on Ulcer patients and
  controls
• Use the data in the table to find the probability
  that a randomly selected patient
• (a) has phenotype A
• (b) is an ulcer patient
• (c) is phenotype O and an ulcer patient
• (d) is phenotype AB or a normal control
• (e) has phenotype A given that the patient is an
  ulcer patient

22
Bayes Theorem

• Often we begin probability analysis with initial
  or prior probabilities.
• Then, from a sample, special report, or a product
  test we obtain some additional information.
• Given this information, we calculate revised or
  posterior probabilities.
• Bayes theorem provides the means for revising
  the prior probabilities.

23
Bayes Theorem

• To find the posterior probability that event Ai
  will occur given that event B has occurred we
  apply Bayes theorem.
• Bayes theorem is applicable when the events for
  which we want to compute posterior probabilities
  are mutually exclusive and their union is the
  entire sample space.

24
Example

• A lab blood test is 95 effective in detecting a
  certain disease when its present. The test also
  yields 1 false positive result. If 0.5
  population has the disease, whats the
  probability that a person with a positive test
  result actually has the disease?