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Copyright by Professor Omar Hasan Kasule Sr
MODULE OUTLINE
2.1 PROBABILITY
2.1.1 Definitions
2.1.2 Classification of Probability
2.1.3 Types of Events
2.1.4 Laws of Probability and Mathematical Properties
2.1.5 Uses of Probability
2.2 VARIABLES
2.2.1 The Random Variable
2.2.2 Common Random Variables
2.2.3 Common Statistical Distributions
2.2.4 Scales of Random Variables
2.2.5 Mathematical Theories of Counting
2.3 THE NORMAL CURVE AND ESTIMATION
2.3.1 Introduction
2.3.2 Properties and Characteristics of the Normal Curve
2.3.3 Use of the Normal Curve for Non-Normal Data
2.3.4 The Z-Score and the Area under the Curve
2.3.5 Estimation
2.4 HYPOTHESES
2.4.1 Hypotheses and the Scientific Method
2.4.2 Null Hypothesis (Ho) & Alternative Hypothesis (Ha):
2.4.3 Hypothesis Testing Using Tests of Significance
2.4.4 Hypothesis Testing Using Confidence Intervals
2.4.5 Conclusions and Interpretations
UNIT 2.1
PROBABILITY
Learning Objectives:
· Probability: definition and types
· Events: mutually exclusive events and independent events
· Laws of probability: additive and multiplicative
· Uses of probability: inference and decision-making
Key Words and Terms:
· Artificial Intelligence
· Chance
· Combinations
· Computer-Assisted Diagnosis
· Decision Support Techniques
· Event, complimentary
· Event, independent
· Event, mutually exclusive
· Gambler's fallacy
· Games theory
· Law, ‘and law
· Law, ‘or’ law
· Law, addition law
· Law, multiplicative law
· Monte Carlo methods
· Permutations
· Probability space
· Probability trees
· Probability, a priori probability
· Probability, Anterior probability
· Probability, probability axioms
· Probability, Bayesian probability
· Probability, classical probability
· Probability, conditional probability
· Probability, empirical probability
· Probability, frequentist probability
· Probability, laws of probability
· Probability, objective probability
· Probability, posterior probability
· Probability, prediction model
· Probability, subjective probability
· Probability, theoretical probability
· Stochastic process
UNIT OUTLINE:
2.1.1 DEFINITIONS
A. Chance, Random and Stochastic Process
B. Probability
C. Tossing a Coin
D. Frequentist Probability
E. Some Terminology
2.1.2 CLASSIFICATION OF PROBABILITY
A. Theoretical Probability and Empirical Probability
B. Subjective Probability & Objective Probability.
C. Bayesian Probability
D. A Priori and A Posteriori Probability:
E. Conditional Probability
2.1.3 TYPES OF EVENTS
A. On the Scale of Exclusion:
B. On the Scale of Dependency:
C. On the Scale of Equal Likelihood
D. On the Scale of Exhaustion
E. Clarifications
2.1.4 LAWS OF PROBABILITY and MATHEMATICAL PROPERTIES
A. Occurrence & Non-Occurrence
B. Additive Law: Occurrence of Any or Both Events
C. Multiplicative Law for Joint Occurrence of Independent Events
D. Multiplication and Addition of Probabilities
E. Mathematical Properties
2.1.5 USES OF PROBABILITY
A. Classical Statistical Inference
B. Bayesian Statistical Inference
C. Bayesian Clinical Decision Making and Epidemiology
D. Use of Probability in Queuing Theories
E. Probability Trees
2.1.1 DEFINITIONS
A. CHANCE, RANDOM and STOCHASTIC PROCESS
Chance is a common language term for apparent lack of cause or design. Chance is measured by probability. Probability was historically used in connection with games of chance but has now found many uses in all branches of science, medicine, research, and business. Chance and random do not mean the same. Random processes lack an aim or purpose. Thus some random phenomena are deliberately made random and are not chance events. The theory of probability is concerned with model ling random phenomena. The formal structure of probability agrees in many ways with basic human intuitive feelings.
A stochastic process arises from chance. A stochastic process refers to variables that change over time according to some random process for example being alive, dying, unemployment, number of children etc. A stochastic process is a random process. A survival process is a type of stochastic process that depends on both the past and the present. A survivor at the present time is one who has survived through previous time periods. A Markov process depends only on the present; the past is irrelevant. Markov processes deal with change of state.
B. PROBABILITY
PROBABILITY AS QUANTIFICATION OF CHANCE
Probability is a measure of likelihood of occurrence of an event. Probability is an intuitive concept that we all employ in ordinary life. Most daily problems involve a probability dimension. The commonsense chance is systematized and quantified in statistical work to make it objective. It is true to say that probability is commonsense reduced to mathematics. Historically probability was used in connection with games of chance. Probability, p, quantifies certainty whereas 1-p quantifies uncertainty. Probability can be expressed as a percentage like 50%, as a chance for example 1 in 4, or as odds for example 80/20 for and 20/80 against.
CONCEPT OF PROBABILITY AND PRE-DETERMINATION
The concept of pure chance is not absolutely true from a tauhidi perspective. All events are pre-destined by Allah. Events in all cases follow natural laws, sunan al llaah, that were fixed by Allah. Probability (theoretical and empirical) is based on sunan. The consistency on which probabilities and predictions are based is because there are underlying sunan.
PROBABILITY and ODDS
Probability measures can legitimately be expressed as odds. The following two expressions show the relation between probability and odds: odds = probability / (1 – probability) and probability = odds / (odds + 1). It is clear from these expressions that as the odds become larger the probability also becomes larger. Whereas the range of probability is from 0 to 1, that of odds is from 0 to infinity. Odds can sometimes be expressed as log odds. The table below shows the approximate correspondence among ranges of probability, odds, and log odds for the same data:
Probability | Odds | Log odds |
0.0 to 0.5 | 0.0 to 0.5 | -¥ to +¥ |
0.5 to 1.0 | 1.0 to +¥ | 0 to +¥ |
It can be seen that probability is symmetric about 0.5 and log odds is symmetric about 0. An odd has no such symmetry.
PROBABILITY MEASURES AND UNCERTAINTY
The concepts of chance or probability are mathematical concepts that manifest the uncertainty and limited knowledge of humans. In our human ignorance we may observe events and conclude that they are random when actually they are following a deterministic order and regularity that are above our understanding. The underlying determinism known only by the ultimate creator of the universe exists and operates.
PROBABILITY AND FIXED PHYSICAL LAWS
Empirical scientific research rests on the bedrock of underlying laws of nature. Science tries to discover these laws and use them. Galileo, Kepler, and Newton were pioneers of a new revolution in science because they understood the role of the laws. Science in Europe before the renaissance could not advance because of superstitious beliefs that unlike the laws of nature could not be the basis for consistent development and growth.
Scientific predictions always involve an element of uncertainty either because of ignorance of all the laws involved or because of imperfect measurement and observation.
Classical physics following Newtonian mechanics was found very useful in description and prediction of motion of large objects. It however broke down when it came to the description of motion of sub-atomic particles for which a new branch of physics, quantum physics, had to be discovered. Motion at the sub-atomical level is more probabilistic and there are greater uncertainties of prediction.
PROBABILITY AND DETERMINISM
Probability, as quantification of chance, conflicts with determinism. The two are however easy to reconcile. All events are deterministic but humans do not know all the factors that go into the determinism. For example it is possible to use mathematical calculations based on physical characteristics of the coin, the manner of tossing, and the environmental conditions to determine for sure whether the toss will come out heads or tails. In the absence of such detailed knowledge, humans use probability (educated guess) to predict the outcome of a toss. Over the long run and with many tosses probability predictions come near the truth because of the underlying deterministic processes. Thus random events are not completely random. There is a hidden structure behind the randomness. Probability theory is an attempt to quantify such factors but in a most indirect way.
PROBABILITY and DISORDER
Probability is used in the detection and description of disorder. It is a dictum of life and science that order can be found in apparent disorder. Investigations such as ECG and EEG are attempts to detect and measure disorder and chaos.
C. TOSSING A COIN
The tossing of a fair coin is used to illustrate probability events. A fair coin is one that is not biased in any way i.e. its results are purely random. The results of tossing a coin constitute a random series. Each throw is considered an independent trial. Each result at each throw, head or tail, is called an event or an outcome. The definition of an event may be more complex when two coins are tossed at the same time. In one such toss there are 4 possible events: head & head, head and tail, tail and head, tail & tail.
There are 2 equally likely outcomes from toss of one coin: Head (H) or tail (T). If two coins are tossed at the same time there are 4 equally likely outcomes: HH, HT, TH, and TT.
D. FREQUENTIST PROBABILITY
Definition and computation of probability as relative frequency is the most popular use of probability. It is the relative frequency of an event on repeated trials under the same conditions. It is in other words the ratio of the occurrence of an event out of all the potentially possible outcomes. For accuracy, all possible trial outcomes must be enumerable leaving nothing out. This can be done manually or empirically by actually carrying out all the trials like tossing a coin. They may also be calculated using the mathematical tools of permutations and combinations. The events or outcomes must all be equally likely. For example getting a head or a tail in a toss of a fair coin is an instance of equally likely outcomes.
The coin tossing experiment illustrates a general principle that can be considered one pf the physical laws. Local disorder may exist with global order. The probability of heads on one toss is less certain than the probability of half the tosses on repeated tossing being heads. There is thus a regular measurable pattern in physical events that becomes clear only on many repetitions of the phenomenon or on examining a large number of observations.
Probability theory is about the global picture and not the local or individual situation. Based on previous data from many people we may conclude that the probability of a death from cancer within 5 years is 30%. This figure does not apply to the individual in a direct way. For an individual the probability is either 100% or 0%. There is for example no scientific answer to the question of a cancer patient ‘why me?’ This is because there are no good methods for computing individual probabilities.
E. SOME TERMINOLOGY
OUTCOME:
The outcomes may be equally likely or not equally likely
The number of successes for event A is denoted as n (A)
PROBABILITY SPACE:
Each possible outcome is called a sample point. The set of all possible outcomes is called the probability space, S.
The probability space may or may not have a definite number of sample points. If the probability space has a definite number of sample points, we denote the number of sample points as n (S).
DEFINITION OF CLASSICAL PROBABILITY
Classical probability of event A is denoted as r (A) and is defined as: r (A) = n (A) / n (S) if the probability space consists of a finite number of equally likely events.
0 =< r =< n
0 =< r/n =< 1) i.e. probability is between 0 and 1
The above can be illustrated on a Venn diagram
ARRANGEMENTS
Special mathematical techniques called arrangements, permutations and combinations, can enable us calculate the probability space theoretically without having to carry out the trials.
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