BIOSTATISTICSTopic: 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

• simulation

repeatedwastrialtimesofnumber

occuredAtimesofnumberAP

)(

Classical approach

• For a given procedure has n different simple events, assuming each of those simple events has an equal chance of occurring

n

s

ntssimple evedifferent ofnumber

occurcanAwaysofnumberAP

)(

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

• Example,105 out of 205 newborn babies are boys, then

P(not boy) = P(girl) =100/205 = 0.488

A

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

)(1)(

)(1)(

1)()(

APAP

APAP

APAP

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

)(

) ()|(

BP

BandAPBAP

Bayes’ theorem

• Dealing with sequential events• Revise a probability value base on additional

information that s later obtained

)|()()|()(

)|()()|(

ABPAPABPAP

ABPAPBAP

)(

) ()|(

BP

BandAPBAP

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

Absolute risk reduction = dc

c

ba

a

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

• Using table 3-4

Absolute risk reduction = Pt / Pc =

dcc

baa

Definitions

• Number needed to treat

= 1 / absolute risk reduction• Rounded up to next larger whole number

Definitions

• Actual odds against event A =– Expressed in the form of m:n

• Actual odds in favor of event A =– Expressed in the form of n:m

)(/)( APAP

)(/)( APAP

Definitions

• Odds ratio =

• Using table 3-4, odds ratio = ad / bc

groupcontrolforeventoffavorinodds

grouptreatmentforeventoffavorinodds

Relative risk versus odds ratio

• Prospective study: relative risk, odds ratio • Retrospective study: odds ratio only

Definitions

• Rate =

– 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

kb

a)(

Counting rule

• For a sequence of 2 events, first event can occur m ways and the second can occur n ways, the events together can occur a total of m*n ways

x~

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

• If there is n items with some items are identical to others, number of permutation is

)!(

!

rn

nprn

!!!

!

21 knnn

n

Combination rule

• Number of combination of r items selected from n different items

!)!(

!

rrn

nCrn