BIOSTATISTICS
Topic: Probability
郭士逢
輔大生科系
2007
Note: These slides are made for teaching purpose only, with contents from the
textbook, Biostatistics for the biological and health science, by M.M. Triola and M.F.
Triola), and supplemental materials published by Pearson Education, Inc. in 2006
Definition
• Event
– A collection of outcomes of a procedure
• Simple event
– An outcome or an event that can not be
further broken down
• Sample space
– For a procedure consist of all possible
simple events
Example - Gender of baby
• Procedure: 1 birth
– Event: female
– Sample space: {male, female}
• Procedure: 3 births
– Event: 2 females and a male
– Sample space:
{fff, ffm, fmf, fmm, mff, mfm, mmf, mmm}
Notation of probabilities
• P denotes a probability
• A, B, C denote specific event
• P(A) denotes the probability of event A
occuring
Defined the probability of an event
• Relative frequency approximation
• Classical approach for equally likely
outcomes
• Subjective probabilities
Relative frequency approximation
• Conduct and observe a procedure and count
the number of time that event A actually
occurs
• Base on the result, P(A)s estimated as
number of times A occured
P( A) 
number of times trial was repeated
• simulation
Classical approach
• For a given procedure has n different simple
events, assuming each of those simple
events has an equal chance of occurring
number of ways A can occur
s
P( A) 

number of different simple events n
Subjective probabilities
• probability of event A s estimated by using
knowledge of relevant circumstance
Law of large number
• As a procedure is repeated again and again,
the relative frequency probability of an event
tends to approach the actual probability
Rules of probability
• The probability of an impossible event is 0
• The probability of an event that is certain to
occur is 1
• For any event, the probability of A is between
0 and 1 inclusive
Complementary event
• Consist of all outcomes in which event A does
not occur, and denoted by A
• Example,105 out of 205 newborn babies are
boys, then
P(not boy) = P(girl) =100/205 = 0.488
Rounding off probabilities
• Express probability in fraction or decimal
rounding off to 3 significant digits
Compound event
• Any event combining two or more simple
events
Addition rule
• P(A or B) = P(in a single trial, event A occur s
or event B occurs or they both occur)
• P(A+B) = P(A) + P(B) – P(A and B)
• Adding n such a way that every outcomes s
counted only once
Definition
• Event A and B are disjoint or mutually
exclusive, if they can not occur at the same
time
Rules of complementary events
P( A)  P( A )  1
P( A)  1  P( A )
P( A )  1  P( A)
Multiplication rule
• P(A and B)=P(event A occurs in a first trial
and event B in a second trial)
Conditional probability
• P(B|A) represents the probability of event B
occurring after event has already occurred
Definition
• Event A and B are independent, if occurrence
of one does not affect the probability of the
occurrence of the other
Formal multiplication rule
• P(A and B) = P(A) . P(B|A)
Probability of at least one
• “At least one” is equivalent to “one or more”
• The complement of getting at least one of a
particular is get no item of that type
Conditional probability
• Conditional probability of an event is the
probability with additional information that
some other event has already occurred
P( A and B)
P( A | B) 
P( B)
Bayes’ theorem
• Dealing with sequential events
• Revise a probability value base on additional
information that s later obtained
P( A and B)
P( A | B) 
P( B)
P( A)  P( B | A)
P( A | B) 
P( A)  P( B | A)  P( A )  P( B | A )
Definitions
• A prior probability is an initial probability
obtained before any additional information
• A posterior probability is a probability that has
been revised by using additional information
that is later obtained
Definitions
• Absolute risk reduction = | P(event occurring
in treatment group ) – P(event occurring in
control group ) |
• From table 3-4
a
c

Absolute risk reduction =
ab
cd
Definitions
• Relative risk is the ratio Pt / Pc
– Pt is the proportion of the characteristic in
treatment group
– Pc is the proportion in control group
a
• Using table 3-4
ab
Absolute risk reduction = Pt / Pc =
c
cd
Definitions
• Number needed to treat
= 1 / absolute risk reduction
• Rounded up to next larger whole number
Definitions
• Actual odds against event A = P( A ) / P( A)
– Expressed in the form of m:n
• Actual odds in favor of event A = P( A) / P( A )
– Expressed in the form of n:m
Definitions
• Odds ratio =
odds in favor of event for treatment group
odds in favor of event for control group
• Using table 3-4, odds ratio = ad / bc
Relative risk versus odds ratio
• Prospective study: relative risk, odds ratio
• Retrospective study: odds ratio only
Definitions
a
• Rate = ( ) k
b
– a = frequency count of the number of people that
event occurred
– b = total number of people exposed to the risk of
the event occurring
– k = multiplier number
Counting rule
• For a sequence of 2 events, first event can
occur m ways and the second can ~
xoccur n
ways, the events together can occur a total of
m*n ways
Factorial
• Denotes the product of decreasing positive
whole number
• Example, 4! = 4*3*2*1 =24
Permutation rule
• Select r items from n available items
n!
n pr 
(n  r )!
• If there is n items with some items are
identical to others, number of permutation is
n!
n1!n2 !  nk !
Combination rule
• Number of combination of r items selected
from n different items
n!
n Cr 
(n  r )! r!