Probability, population and sample

Basic concepts: Probability, Population & Sample
Dr. S. A. Rizwan, M.D.
Public Health Specialist
SBCM, Joint Program – Riyadh
Ministry of Health, Kingdom of Saudi Arabia

Learning objectives
Demystifying statistics! – Lecture 1 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Familiarise with terms used in probability
• Define probability
• Describe the 3 approaches of probability
• Understand the basic laws of probability
• Solve problems based on above laws
• Describe the concepts of Population and Sample
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Section 1:
Probability basics
Demystifying statistics! – Lecture 1 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh 3

A few important terms
• Random experiment
• Sample space
• Event
• exhaustive, impossible, elementary, composite,
certain, mutually exclusive, independent,
dependent, favourable, equally likely
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A few important terms
Event type Example
Independent Two seeds are sown, germination of one is
not affected by the other.
Dependent If we draw a card from a pack of well
shuffled cards, if the first card drawn is not
replaced then the second draw is
dependent on the first draw.
Mutually
exclusive
In observation of seed germination the
seed may either germinate or it will not
germinate.
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Probability – what is it?
• Chance that something will happen, how likely an event will happen
• Can be measured with a number like "10% chance of rain", or using
words like impossible, unlikely, possible, even chance, likely and certain
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Probability – what is it?
• Example of a fair die
• P (landing a 5)
• P (landing an even number)
• Example of marbles
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Probability – approaches to calculate
• Theoreticalprobability
• Relative frequency
• Subjective probability
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Laws of Probability – Set analogy
• Union
• Intersection
• Complement
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Thought exercises:
Example 1. A bag contains 5 red marbles, 3 blue marbles, and 2 green
marbles.
Q1. pr (red) + pr (blue) + pr (green)
Q2. pr (red) + pr (not red)
Q3. pr (red or green)
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Thought exercises:
Example 2. Roll a die and flip a coin.
Q1. pr (heads and roll a 3)
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Thought exercises:
Example 3. There are 20 people in the room: 12 girls (5 with blond hair and
7 with brown hair) and 8 boys (4 with blond hair and 4 with brown hair).
There are a total of 9 blonds and 11 with brown hair.
Q1. pr (girl or blond)
Q2. pr (girl with brown hair) and pr (girl)
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Thought exercises:
Example 4. toss a coin 4 times
Q1. What is the probability of getting at least one head on the 4 tosses
pr (at least one H)
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Thought exercises:
Example 5. An individual applying to a college determines that he has an
80% chance of being accepted, and he knows that dormitory housing will
only be provided for 60% of accepted students.
Q1. What is the chance of the student being accepted and receiving
dormitory housing?
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Thought exercises:
Example 6. A = “Patient has liver disease.” Past data tells you that 10% of
patients entering your clinic have liver disease. B = “Patient is an alcoholic.”
Five percent of the clinic’s patients are alcoholics. Among patients
diagnosed with liver disease, 7% are alcoholics.
Q1. Find out a patient’s probability of having liver disease if they are an
alcoholic.
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Laws of Probability - 1
• The probability of an event is
between 0 and 1
• A probability of 1 is equivalent
to 100% certainty
• Probabilities can be expressed
at fractions, decimals, or
percentage
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• The sum of the probabilities of all possible outcomes is 1 or 100%.
• If A, B, and C are the only possible outcomes,
• then p(A) + p(B) + p(C) = 1
Example: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles.
p (red) + p (blue) + p (green) = 1
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• The sum of the probability of an event occurring and it not occurring is 1.
• p(A) + p(not A) = 1
• p(not A) = 1 - p(A)
p (red) + p (not red) = 1
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• If two events A and B are independent
• then the probability of A and B occurring is the product of their
individual probabilities.
• p (A and B) = p (A) X p (B)
Example: roll a die and flip a coin. p (heads and roll a 3) = p (H) X p (3)
*Multiplicative Theorem
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Laws of Probability - 4a
• If two events A and B are dependent
• then the probability of A and B occurring is
• p (A and B) = p (B|A) X p (A) = p (A|B) X p (B)
• An individual applying to a college determines that he has an 80% chance of being
accepted, and he knows that dormitory housing will only be provided for 60% of
accepted students.
• What is the chance of the student being accepted and receiving dormitory housing?
• P(Accepted and Dormitory Housing)
• = P(Dormitory Housing | Accepted) * P(Accepted)
• = (0.60)*(0.80) = 0.48
*Multiplicative Theorem
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• If two events A and B are mutually exclusive
• then the probability of A or B occurring is the sum of their individual
probabilities.
• p (A or B) = p (A) + p (B)
p (red or green) = p (red) + p (green)
*Addition Theorem
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• If two events A and B are NOT mutually exclusive
• then the probability of A or B occurring is the sum of their individual
probabilities minus the probability of both A and B occurring
• p (A or B) = p (A) + p (B) – p (A and B)
Example: 12 girls (5 with blond hair and 7 with brown hair) and 8 boys (4
with blond hair and 4 with brown hair). There are a total of 9 blonds and 11
with brown hair. One person from the group is chosen randomly.
p (girl or blond) = p (girl) + p (blond) – p (girl and blond)
*Addition Theorem
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• The probability of at least one event occurring out of multiple events is
equal to one minus the probability of none of the events occurring.
• p (at least one) = 1 – p (none)
Example: What is the probability of getting at least one head on the 4 throw
of a coin?
p (at least one H) = 1 – p (no H) = 1 – p (TTTT)
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• If event B is a subset of event A,
• then the probability of B is less than or equal to the probability of A.
• p (B) ≤ p (A)
Example: There are 20 people in the room: 12 girls (5 with blond hair and 7
with brown hair) and 8 boys (4 with blond hair and 4 with brown hair).
p (girl with brown hair) ≤ p (girl)
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Conditional Probability
• Two events A and B are said to be dependent, when B can occur only
when A is known to have occurred (or vice versa)
• The probability attached to such an event is called the conditional
probability and is denoted by P (A/B)
*Bayes Theorem
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Conditional Probability
*Bayes Theorem
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Conditional Probability - Example
• A = “Patient has liver disease.” Past data tells you that 10% of patients
entering your clinic have liver disease. P(A) = 0.10.
• B = “Patient is an alcoholic.” Five percent of the clinic’s patients are
alcoholics. P(B) = 0.05.
• Among patients diagnosed with liver disease, 7% are alcoholics. This is
your B|A.
• Finding out a patient’s probability of having liver disease if they are an
alcoholic.
• P(A|B) = (0.07 * 0.1) / 0.05 = 0.14
*Bayes Theorem
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Section 2:
Population & Sample

Population
• The population is the set of entities under study
• A population includes all of the elements from a set of data.
• A measurable characteristic of a population - parameter

Sample
• A sample consists of one or more observations from the population
• A measurable characteristic of a sample is called a statistic.

Sample

Descriptive statistics
• Descriptive statistics provide a concise summary of data.
• You can summarize data numerically or graphically

Descriptive statistics

Inferential statistics
• Inferential statistics use a random sample of data taken from a population
to describe and make inferences about the population.

Inferential statistics

Take home messages
• Understanding probability and its laws
are important to understanding
biostatistics
• The concepts of population and sample
are essential for understanding inferential
statistics
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Thank you!
Email your queries to sarizwan1986@outlook.com
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Probability, population and sample

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