M RABEEK .S

1st M.Sc Microbiology

PROBABILITY & PROBABILITY DISTRIBUTION

INTRODUCTION

• Probability is the likehood or chance that a

particular event will or will not occur.

• The theory of probability provides a quantitative

measure of uncertainty of occurrence of different

events resulting from a random experiment, in

terms of quantitative measures ranging from 0 to 1

EXPERIMENT:

it is a process which produces outcomes.

Example:

Tossing a coin is an experiment. An interview

to gauge the job satisfaction levels of the employees

in an organization is an experiment.

SOME COMMON PROBABILITY PARLANCE IN

MEDICAL:

• A patient has a 50-50 chance of surviving a

certain operation.

• Another physician may say that she is 95%

certain that a patient has a particular disease.

• A public health nurse may say that nine times

out of ten a certain client will break an

appointment.

CONT….

• We measure the probability of occurrence of some event

by a number between zero and one .

• The more likely the event, the closer the prob. is to one and more unlikely it is the closer the prob. to ‘0’.

• An event that can’t occur has a probability zero, and an event that is certain to occur has a probability of one.

• Health sciences researchers continually ask themselves if the result of their efforts could have occurred by chance alone or if some other force was operating to produce the observed effect

TWO VIEWS OF PROBABILITY:

• Objective

• subjective

• Objective Probability:

• Classical or a prior probability

• The relative frequency or a posterior

CLASSICAL PROBABILITY

• Definition-

• If an event can occur in N mutually exclusive and equally likely ways, and if ‘m’ of these possess a trait, E, the probability of the occurrence of E is equal to m/N.

• Symbolically-

•

• P(E) – Prob. of E =

P(E)=

m

N

CONT…

• If the process is repeated a large number of times, n ,

and if some resulting event with the characteristic E

occurs ‘m’ times, the relative frequency of occurrence

of E, m/n, will be approximately equal to the

probability of E. Symbolically P(E) = m/n

• Subjective Probability:

• Probability measures the confidence that a particular

individual has in the truth of a particular proposition.

CONT…

• This concept doesn’t rely on the repeatability of any

process.

• In fact, by applying this concept of probability, one

may evaluate the probability of an event that can only

happen once, for example, the probability that a cure

for cancer will be discovered within the next 10 years

LAWS OF PROBABILITY

• For any event probability lies between 0 & 1

• It is represented in percentages, ratios, fractions

• Each event has a complementary event

i.e. P(E1) + P’(E1) =1

TYPES OF PROBABILITY

• Marginal Probability;

• Union Probability;

• Joint Probability;

• Conditional Probability.

MARGINAL PROBABILITY

• It is the first type of probability.

• A marginal or unconditional probability is the simple

probability of the occurrence of an event.

• Denoted by P(E) where ‘E’ is some event.

P(E)= Number of outcomes favorable to occurrence of E

Total number of outcomes

UNION PROBABILITY

• Second type of probability.

• If E1 & E2 are two Events, then Union probability isdenoted by P(E1 U E2 ).

• It is the probability that Event E1 will occur or that EventE2 will occur or both Event E1 & Event E2 will occur.

• For example, union probability is the probability that aperson either owns a Maruti 800 or Maruti Zen. Forqualifying to be part of the union, a person has to haveatleast one of these cars.

JOINT PROBABILITY

• It is the third type of probability

• If E1 & E2 are two Events, then Joint probability is

denoted by P(E1∏E2 ).

• It is the probability of the occurrence of Event E1 and

Event E2.

• For example, it is the probability that a persons owns both

a Maruti 800 & Maruti Zen; for joint probability, owning a

single car is not sufficient.

CONDITIONAL PROBABILITY

• It is the fourth type of probability.

• Conditional Probability of two Events E1 & E2 is

generally denoted by P(E1/E2).

• It is probability of the occurrence of E1 given that E2 has

already occurred.

• Conditional probability is the probability that a person

owns a Maruti 800 given that he already has a Maruti Zen.

APPLICATION OF PROBABILITY:

• Probability theory is applied in day to day life in risk

assessments and in trade on financial markets.

• Another Significant application of probability theory in

everyday life is reliability. Many consumer products, such

as automobiles and consumer electronics use reliability

theory in product design to reduce the probability of

failure.

PROBABILITY DISTRIBUTION

DISCRETE & CONTINUOUS RANDOM

DISTRIBUTIONS

• A random variable is a variable which contains theoutcome of a chance experiment; for example, in anexperiment to measure the number of customers whoarrive in a shop during a time interval of 2 minutes; thepossible outcome may vary from 0 to n customers; theseoutcomes (0,1,2,3,4,…n)are the values of the randomvariable.

• These random variables are called discrete randomvariables

In other words , a random variable which assumes either a

finite number of values or a countable infinite number of

possible values is termed as Discrete Random variable.

On the other hand, random variables that assumes any

numerical value in an interval or can take values at every point

in a given interval is called continuous random variable. For

example, temperatures recorded for a particular city can

assume any number like 32O F, 32.5O F 35.8O F

Experiment outcomes which are based on measurement scale

such as time, distance, weight & temperature can be explained

by Continuous Random variable.

Con

BINOMIAL PROBABILITY DISTRIBUTION

• Most commonly used & widely known distribution among

all discrete distributions.

• It is a sequence of repeated trials, called Bernoulli

Process which is characterized by:

1. Only two mutually exclusive outcomes are possible;( one

is referred to as success & the other as failure).

2. The outcomes in a series of trials/observation constitute

independent events.

3. Probability of success (p) or failure (q) is constant over a

number of trials.

4. The number of events is discrete & can be represented by

integers(0,1,2,3,4,onwards).

POISSON DISTRIBUTION

• It is named after the famous French MathematicianSimeon Poisson.

• It is also a discrete distribution; but there are a fewdifferences between Binomial & Poisson distributions. Fora given number of trials the binomial distributiondescribes a distribution of two possible outcomes: eithersuccess or failure whereas Poisson focuses on the numberof discrete occurrences over an interval.

• It is widely used in the field of managerial decisionmaking widely used in queuing models.

POISSON PROCESS CONDITIONS

• The event occur in a continuum of time & at a randomly

selected point & event either occurs or doesn’t occur.

• Whether the event occur or doesn’t occur at a point, it is

independent of the previous point where the event may

have occurred or not.

• The probability of occurrence of events remains

same/constant over the whole period or throughout the

continuum.

NORMAL DISTRIBUTION

• It is the most commonly used distribution among all

probability distributions.

• It has a wide range of practical application example,

where the random variables are human characteristics such

as height, weight, speed, IQ scores.

• Normal distribution was invented in the 18th century.