University of Gondar
College of Medicine and Health Sciences
Institute of Public Health
Department of Epidemiology and Biostatistics
Chapter Four: Probability and probability distributions
Saturday, July 6, 2024
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Probability and Probability Distributions, Schemes and Mind Maps of Object Oriented Analysis and Design
A comprehensive overview of probability concepts, including basic terms, types of probability, set theory, and probability distributions. It covers key topics such as classical and empirical probability, random variables, common probability distributions (binomial, poisson, normal, and t-distribution), and their properties. The document aims to lay the foundation for understanding statistical inference and is intended for students studying epidemiology and biostatistics. It includes learning outcomes, examples, and exercises to reinforce the concepts presented. The content is relevant for university-level courses in statistics, data analysis, and research methods, particularly in the fields of public health, medicine, and health sciences.
Typology: Schemes and Mind Maps
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University of Gondar
College of Medicine and Health Sciences
Institute of Public Health
Department of Epidemiology and Biostatistics
Chapter Four : Probability and probability distributions
Saturday, July 6, 2024
Learning outcomes After studying this chapter, the student will be able to: 4.1 Define basic terms in probability 4.2 Describe set theory and probability 4.3 Identify types of probability 4.4 Identify types of random variable and probability distribution 4.5 List common probability distributions and their properties
Basic Terms of Probability
- (^) Probability can be defined as the chance of an event
occurring.
- (^) Probability experiment: is a process that leads to well- defined results or is an action through which specific results/outcomes (counts, measurements or responses) are obtained. But that is the result cannot be predicted. Example:
- (^) Tossing a coin and observing the face showing up is a probability experiment.
- (^) Outcome: It is the result of a single trial in a probability experiment. It is also called simple event. Example: the outcome of the sex of a newborn from a mother in delivery room is either Male or female
Saturday, July 6, 2024 Wullo S. Basic concepts con'td….
- (^) Sample space: Each conceivable outcome of an experiment is called a sample point. The totality of all sample points is called a sample space and i s denoted by S.
- (^) Event: An outcome or a combination of outcomes of a random experiment is called an event. It is a subset of the sample space of a random experiment.
- (^) Equally-likely Approach: If an experiment must result in n equally likely outcomes, then each possible outcome must have probability 1/n of occurring.
- (^) Mutually exclusive events: when the occurrence of any one event excludes the occurrence of the other event. Mutually exclusive events cannot occur simultaneously. 5
Exercise
Find the sample space for the gender of the children
if a family has three children. Use B for boy and G
for girl
And also find:
a. The probability of obtaining at least two girls in a family? b. The probability of getting at most two boys in a family? c. The probability of getting one boys and two girls in a family?
Types of probability
- Classical (or theoretical) probability (^) It is used when each outcome in a sample space is equally likely to occur. (^) That is if an experiment has n equally likely outcomes, then each possible outcome must have probability of 1/n to occur Or, equivalently the probability for event E is; Example: The probability of getting at least one female birth from two pregnant mothers is: ¾ = 0.
Example 2
In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities a. A person has type O blood b. A person has type A or type B blood c. A person has neither type A nor type O blood d. A person does not have type AB blood
Solution
(^) P(o) = 21/50 = 0. (^) P(A)= 22/50 = 0. (^) P (A or B)=p(A)+P(B)= 22/50+5/50=27/ (^) Do others in this way? Blood type Frequency A 22 B 5 AB 2 O 21 Total 50
Staff Gender Male Female Total Physician 2 3 5 Nurse 1 7 8 Total 3 10 13
Saturday, July 6, 2024 Wullo S.
- (^) In this view, probability is treated as a quantifiable level of belief ranging from 0 (complete disbelief) to 1 (complete belief)
- (^) For instance, an experienced physician may say “this patient has a 50% chance of recovery.
- (^) All probabilities are a type of relative frequency—the number of times something can occur divided by the total number of possibilities or occurrences.
3. Subjective probability
14
Introduction to expectation
Definition : the expected value (also known as the
mean) of a random variable is a measure of the
center location for the random variable.
1. Discrete R.V
E(X) = X
1
P(X
1
) +X
2
P(X
2
) +…. +X
n
P(X
n
2. Continuous R.V
^ ^ n i Xi P X i 1 . E X X. f ( x ) d ( x ) b a
Variance Probability distribution
- (^) The expected value of X is its mean
Mean of X= E(X)
- (^) The variance of X is given by: Variance of X=Var(x) = ^ ^ ^ ^ 2 2 E X ( E X ) X f x d x if X is continuous E X X P X if X is discrete x i n i i ( ) ( ). 2 1 2 2
1. Binomial Distribution A binomial experiment is a probability experiment that satisfies the following four requirements called assumptions of a binomial distribution.
- (^) The experiment consists of n identical trials.
- (^) Each trial has only one of the two possible mutually exclusive outcomes, success or a failure.
- (^) The probability of success does not change from trial to trial, and
- (^) The trials are independent, thus we must sample with replacement
Binomial distribution Cont..