Draw the figure. We want the area between -2.55 and +2.55.
Look up the area to the left of -2.55 in the table. It is 0.0049. Also look up
the area to the left of +2.55. It is 0.9937. Subtract these two areas. 0.9937
- 0.0049 = 0.9888. Hence, the probability that a z picked at random will
have a value between -2.55 and +2.55 is 0.9888 or 98.88%
It is a consolidation of basic probability concepts worth understanding before attempting to apply probability concepts for predictions. The material is formed from different sources. ll the sources are acknowledged.
1 Probability Please read sections 3.1 – 3.3 in your .docxaryan532920
1
Probability
Please read sections 3.1 – 3.3 in your textbook
Def: An experiment is a process by which observations are generated.
Def: A variable is a quantity that is observed in the experiment.
Def: The sample space (S) for an experiment is the set of all possible outcomes.
Def: An event E is a subset of a sample space. It provides the collection of outcomes
that correspond to some classification.
Example:
Note: A sample space does not have to be finite.
Example: Pick any positive integer. The sample space is countably infinite.
A discrete sample space is one with a finite number of elements, { }1,2,3,4,5,6 or one that
has a countably infinite number of elements { }1,3,5,7,... .
A continuous sample space consists of elements forming a continuum. { }x / 2 x 5< <
2
A Venn diagram is used to show relationships between events.
A intersection B = (A ∩ B) = A and B
The outcomes in (A intersection B) belong to set A as well as to set B.
A union B = (A U B) = A alone or B alone or both
Union Formula
For any events A, B, P (A or B) = P (A) + P (B) – P (A intersection B) i.e.
P (A U B) = P (A) + P (B) – P (A ∩ B)
3
cA complement not A A ' A A = = = =
A complement consists of all outcomes outside of A.
Note: P (not A) = 1 – P (A)
Def: Two events are mutually exclusive (disjoint, incompatible) if they do not intersect,
i.e. if they do not occur at the same time. They have no outcomes in common.
When A and B are mutually exclusive, (A ∩ B) = null set = Ø, and P (A and B) = 0.
Thus, when A and B are mutually exclusive, P (A or B) = P (A) + P (B)
(This is exactly the same statement as rule 3 below)
Axioms of Probability
Def: A probability function p is a rule for calculating the probability of an event. The
function p satisfies 3 conditions:
1) 0 ≤ P (A) ≤1, for all events A in the sample space S
2) P (Sample Space S) = 1
3) If A, B, C are mutually exclusive events in the sample space S, then
P(A B C) P(A) P(B) P(C)∪ ∪ = + +
4
The Classical Probability Concept: If there are n equally likely possibilities, of which one
must occur and s are regarded as successes, then the probability of success is s
n
.
Example:
Frequency interpretation of Probability: The probability of an event E is the proportion of
times the event occurs during a long run of repeated experiments.
Example:
Def: A set function assigns a non-negative value to a set.
Ex: N (A) is a set function whose value is the number of elements in A.
Def: An additive set function f is a function for which f (A U B) = f (A) + f (B) when A and
B are mutually exclusive.
N (A) is an additive set function.
Ex: Toss 2 fair dice. Let A be the event that the sum on the two dice is 5. Let B be the
event that the sum on ...
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 4: Probability
4.1: Basic Concepts of Probability
It is a consolidation of basic probability concepts worth understanding before attempting to apply probability concepts for predictions. The material is formed from different sources. ll the sources are acknowledged.
1 Probability Please read sections 3.1 – 3.3 in your .docxaryan532920
1
Probability
Please read sections 3.1 – 3.3 in your textbook
Def: An experiment is a process by which observations are generated.
Def: A variable is a quantity that is observed in the experiment.
Def: The sample space (S) for an experiment is the set of all possible outcomes.
Def: An event E is a subset of a sample space. It provides the collection of outcomes
that correspond to some classification.
Example:
Note: A sample space does not have to be finite.
Example: Pick any positive integer. The sample space is countably infinite.
A discrete sample space is one with a finite number of elements, { }1,2,3,4,5,6 or one that
has a countably infinite number of elements { }1,3,5,7,... .
A continuous sample space consists of elements forming a continuum. { }x / 2 x 5< <
2
A Venn diagram is used to show relationships between events.
A intersection B = (A ∩ B) = A and B
The outcomes in (A intersection B) belong to set A as well as to set B.
A union B = (A U B) = A alone or B alone or both
Union Formula
For any events A, B, P (A or B) = P (A) + P (B) – P (A intersection B) i.e.
P (A U B) = P (A) + P (B) – P (A ∩ B)
3
cA complement not A A ' A A = = = =
A complement consists of all outcomes outside of A.
Note: P (not A) = 1 – P (A)
Def: Two events are mutually exclusive (disjoint, incompatible) if they do not intersect,
i.e. if they do not occur at the same time. They have no outcomes in common.
When A and B are mutually exclusive, (A ∩ B) = null set = Ø, and P (A and B) = 0.
Thus, when A and B are mutually exclusive, P (A or B) = P (A) + P (B)
(This is exactly the same statement as rule 3 below)
Axioms of Probability
Def: A probability function p is a rule for calculating the probability of an event. The
function p satisfies 3 conditions:
1) 0 ≤ P (A) ≤1, for all events A in the sample space S
2) P (Sample Space S) = 1
3) If A, B, C are mutually exclusive events in the sample space S, then
P(A B C) P(A) P(B) P(C)∪ ∪ = + +
4
The Classical Probability Concept: If there are n equally likely possibilities, of which one
must occur and s are regarded as successes, then the probability of success is s
n
.
Example:
Frequency interpretation of Probability: The probability of an event E is the proportion of
times the event occurs during a long run of repeated experiments.
Example:
Def: A set function assigns a non-negative value to a set.
Ex: N (A) is a set function whose value is the number of elements in A.
Def: An additive set function f is a function for which f (A U B) = f (A) + f (B) when A and
B are mutually exclusive.
N (A) is an additive set function.
Ex: Toss 2 fair dice. Let A be the event that the sum on the two dice is 5. Let B be the
event that the sum on ...
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 4: Probability
4.1: Basic Concepts of Probability
Similar to 4Probability and probability distributions (1).pptx (20)
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2. Chapter 34, Ganong’s Review of Medical Physiology, 26th edition
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4. Non-respiratory functions of the lungs https://academic.oup.com/bjaed/article/13/3/98/278874
Ethanol (CH3CH2OH), or beverage alcohol, is a two-carbon alcohol
that is rapidly distributed in the body and brain. Ethanol alters many
neurochemical systems and has rewarding and addictive properties. It
is the oldest recreational drug and likely contributes to more morbidity,
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5th edition of the Diagnostic and Statistical Manual of Mental Disorders
(DSM-5) integrates alcohol abuse and alcohol dependence into a single
disorder called alcohol use disorder (AUD), with mild, moderate,
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In the DSM-5, all types of substance abuse and dependence have been
combined into a single substance use disorder (SUD) on a continuum
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the 11 DSM-5 behaviors be present within a 12-month period (mild
AUD: 2–3 criteria; moderate AUD: 4–5 criteria; severe AUD: 6–11 criteria).
The four main behavioral effects of AUD are impaired control over
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Tom Selleck Health: A Comprehensive Look at the Iconic Actor’s Wellness Journeygreendigital
Tom Selleck, an enduring figure in Hollywood. has captivated audiences for decades with his rugged charm, iconic moustache. and memorable roles in television and film. From his breakout role as Thomas Magnum in Magnum P.I. to his current portrayal of Frank Reagan in Blue Bloods. Selleck's career has spanned over 50 years. But beyond his professional achievements. fans have often been curious about Tom Selleck Health. especially as he has aged in the public eye.
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Introduction
Many have been interested in Tom Selleck health. not only because of his enduring presence on screen but also because of the challenges. and lifestyle choices he has faced and made over the years. This article delves into the various aspects of Tom Selleck health. exploring his fitness regimen, diet, mental health. and the challenges he has encountered as he ages. We'll look at how he maintains his well-being. the health issues he has faced, and his approach to ageing .
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4Probability and probability distributions (1).pptx
1. 4.INTRODUCTION TO PROBABILITY & PROBABILITY DISTRIBUTIONS
7/8/2023 Probability & probability distributions 1
Emiru Merdassa
2. Definition
Probability: the chance that an uncertain
event will occur (always between 0 and 1)
7/8/2023 Probability & probability distributions
2
0 ≤ P(A) ≤ 1 For any event A
Certain
Impossible
0.5
1
0
Example:
• Physician may say that a patient has a 50-50 chance of
surviving a certain operation.
• 95% certain that a patient has a particular disease.
Most people express probabilities in terms of
percentages.
3. Probability terms
•Outcomes: results of each trial
•Sample space: set of all possible outcome
•Sample points: elements of the sample space of
outcome
•Event: subset of the sample space
7/8/2023 Probability & probability distributions 3
(continued)
4. Important terms
• Intersection of Events – If A and B are two events in a sample
space S, then the intersection, A ∩ B, is the set of all outcomes
in S that belong to both A and B
7/8/2023 Probability & probability distributions 4
(continued)
A B
AB
S
5. Important terms
• Two events A and B are mutually exclusive if they cannot both
happen at the same time
• P (A ∩ B) = 0
7/8/2023 Probability & probability distributions
5
(continued)
A B
S
6. Important terms
• Union of Events – If A and B are two events in a sample space S,
then the union, A U B, is the set of all outcomes in S that
belong to either A or B
7/8/2023 Probability & probability distributions 6
(continued)
A B
The entire shaded
area represents
A U B
S
7. Important terms
• Events E1, E2, … Ek are Collectively Exhaustive events if E1 U E2 U
. . . U Ek = S
• i.e., the events completely cover the sample space
• The Complement of an event A is the set of all basic outcomes
in the sample space that do not belong to A. The complement
is denoted as A
7/8/2023 Probability & probability distributions
7
(continued)
A
S
A
8. 7/8/2023 Probability & probability distributions 8
Example
Let the Sample Space be the collection of all possible
outcomes of rolling one die:
S = [1, 2, 3, 4, 5, 6]
Let A be the event “Number rolled is even”
Let B be the event “Number rolled is at least 4”
Then
A = [2, 4, 6] and B = [4, 5, 6]
9. Example
7/8/2023 Probability & probability distributions 9
(continued)
S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]
5]
3,
[1,
A
6]
[4,
B
A
6]
5,
4,
[2,
B
A
S
6]
5,
4,
3,
2,
[1,
A
A
Complements:
Intersections:
Unions:
[5]
B
A
3]
2,
[1,
B
10. Example
Mutually exclusive:
oA and B are not mutually exclusive
• The outcomes 4 and 6 are common to both
Collectively exhaustive:
oA and B are not collectively exhaustive
• A U B does not contain 1 or 3
7/8/2023 Probability & probability distributions
10
(continued)
S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]
11. Independent Events
• Two events A and B are independent if the probability of
the first one happening is the same no matter how the
second one turns out or
• The outcome of one event has no effect on the occurrence
or non-occurrence of the other.
P(A∩B) = P(A) x P(B) (Independent events)
P(A∩B) ≠ P(A) x P(B) (Dependent events)
Example:
OThe outcomes on the first and second coin tosses are
independent
7/8/2023 Probability & probability distributions 11
12. Ctd…
Two categories of probability
1. Objective probability and
2. Subjective probability
1) Objective probability
a) Classical probability &
b) Relative frequency probability.
7/8/2023 Probability & probability distributions 12
13. A) Classical Probability
• Is based on gambling ideas
• Example : Rolling a die
• There are 6 possible outcomes:
• Sample space = {1, 2, 3, 4, 5, 6}.
• Each is equally likely
• P(i) = 1/6, i=1,2,...,6.
→ P(1) = 1/6
→ P(2) = 1/6
…
→ P(6) = 1/6
SUM = 1
7/8/2023 Probability & probability distributions 13
14. B) Relative Frequency Probability
• In the long run process.
• The proportion of times the event A occurs —in a large number of
trials repeated under essentially identical conditions
Definition:
• If a process is repeated a large number of times (n), and if an event
with the characteristic E occurs m times, the relative frequency of E,
Probability of E = P(E) = m/n.
7/8/2023 Probability & probability distributions 14
15. Example
7/8/2023 Probability & probability distributions 15
•If you toss a coin 100 times and head comes up 40 times,
P(H) = 40/100 = 0.4.
•If we toss a coin 10,000 times and the head comes up
5562,
P(H) = 0.5562.
•Therefore, the longer the series and the longer sample
size, the closer the estimate to the true value.
16. Relative probability
•Since trials cannot be repeated an infinite number of
times, theoretical probabilities are often estimated by
empirical probabilities based on a finite amount of
data
Example:
Of 158 people who attended a dinner party, 99 were ill.
P (Illness) = 99/158 = 0.63 = 63%.
7/8/2023 Probability & probability distributions 16
(continued)
17. 2. Subjective Probability
• Personalistic (represents one’s degree of belief in the occurrence
of an event).
• E.g., If someone says that he is 95% certain that a cure for AIDS
will be discovered within 5 years, then he means that:
• P(discovery of cure for AIDS within 5 years) = 95% = 0.95
• Although the subjective view of probability has enjoyed
increased attention over the years, it has not fully accepted by
scientists.
7/8/2023 Probability & probability distributions 17
18. Properties of Probability
1. The numerical value of a probability always lies between 0 and 1,
inclusive(0 P(E) 1)
2. The sum of the probabilities of all mutually exclusive outcomes is
equal to 1.
P(E1) + P(E2 ) + .... + P(En ) = 1.
3. For two mutually exclusive events A and B,
P(A or B ) = P(AUB)= P(A) + P(B).
If not mutually exclusive:
P(A or B) = P(A) + P(B) - P(A and B)
7/8/2023 Probability & probability distributions 18
19. Properties of Probability
4. The complement of an event A, denoted by Ā or Ac, is the
event that A does not occur
• Consists of all the outcomes in which event A does NOT
occur: P(Ā) = P(not A) = 1 – P(A)
• Where Ā occurs only when A does not occur.
7/8/2023 Probability & probability distributions 19
20. Basic Probability Rules
1. Addition rule
If events A and B are mutually exclusive:
P(A or B) = P(A) + P(B)
P(A and B) = 0
More generally:
P(A or B) = P(A) + P(B) - P(A and B)
P(event A or event B occurs or they both occur)
7/8/2023 Probability & probability distributions 20
21. Example
The probabilities representing years of schooling
completed by mothers of newborn infants
7/8/2023 Probability & probability distributions 21
Mother’s education level Probability
≤ 8 years 0.056
9 to 11 years 0.159
12 years 0.321
13 to 15 years 0.218
≥16 years 0.230
Not reported 0.016
22. Ctd…
7/8/2023 Probability & probability distributions 22
• What is the probability that a mother has
completed < 12 years of schooling?
P( 8 years) = 0.056 and
P(9-11 years) = 0.159
• Since these two events are mutually exclusive,
P( 8 or 9-11) = P( 8 U 9-11)
= P( 8) + P(9-11)
= 0.056+0.159
= 0.215
23. Exercise
i. Suppose two doctors, A and B, test all patients coming into a clinic for syphilis.
Let events A+ = {doctor A makes a positive diagnosis} and B+ = {doctor B
makes a positive diagnosis}.
ii. Suppose doctor A diagnoses 10% of all patients as positive, doctor B diagnoses
17% of all patients as positive, and both doctors diagnose 8% of all patients as
positive.
Suppose a patient is referred for further lab tests if either doctor A or B makes a
positive diagnosis.
• What is the probability that a patient will be referred for further lab tests(P(Α+
∪
𝐵+
)?
7/8/2023 Probability & probability distributions 23
25. The Normal distribution
•Normal distributions are recognized by their bell shape.
•A large percentage of the curve’s area is located near its
center and that its tails approach the horizontal axis as
asymptotes.
7/8/2023 Probability & probability distributions 25
26. Normal distribution
7/8/2023 Probability & probability distributions 26
Normal distributions are defined by:
, - < x < .
where μ represents the mean of the distribution
and σ represents its standard deviation.
f(x) =
1
2
e
x-
2
1
2
27. There are many different Normal distributions, each
with its own μ and σ, let X ~ N(μ, σ) represent a specific
member of the Normal distribution family.
The symbol “~” is read as “distributed as.”
For example,
X ~ N(100, 15) is read as “X is a Normal random
variable with mean 100 and standard deviation 15.
7/8/2023 Probability & probability distributions 27
28. 7/8/2023 Probability & probability distributions 28
1. The mean µ tells you about location -
• Increase µ - Location shifts right
• Decrease µ – Location shifts left
• Shape is unchanged
2. The variance σ2 tells you about narrowness or flatness of the bell
-
• Increase σ2 - Bell flattens. Extreme values are more likely
• Decrease σ2 - Bell narrows. Extreme values are less likely
• Location is unchanged
31. Properties of the Normal Distribution
1. It is symmetrical about its mean, .
2. The mean, the median and mode are almost equal. It is
unimodal.
3. The total area under the curve about the x-axis is 1
square unit.
4. The curve never touches the x-axis.
5. As the value of increases, the curve becomes more
and more flat and vice versa.
7/8/2023 Probability & probability distributions 31
32. 7/8/2023 Probability & probability distributions 32
6. Perpendiculars of:
± 1SD contain about 68%;
±2 SD contain about 95%;
±3 SD contain about 99.7% of the area under the
curve.
7. The distribution is completely determined by the
parameters and .
34. Standard Normal Distribution
It is a normal distribution that has a mean equal to 0 and a
SD equal to 1, and is denoted by N(0, 1).
The main idea is to standardize all the data that is given by
using Z-scores.
These Z-scores can then be used to find the area
(probability) under the normal curve.
7/8/2023 Probability & probability distributions 34
35. Z - transformation
•If a random variable X~N(,) then we can transform
it to a SND with the help of Z-transformation
Z =
𝑿−
•Z represents the Z-score for a given x value
7/8/2023 Probability & probability distributions 35
36. Consider redefining the scale to be in terms of how many SDs
away from mean for normal distribution, μ=110 and σ = 15.
SDs from mean using
𝒙−𝟏𝟏𝟎
𝟏𝟓
=
𝒙−
7/8/2023 Probability & probability distributions 36
Value x
50 65 80 95 110 125 140 155 171
-4 -3 -2 -1 0 1 2 3 4
37. 7/8/2023 Probability & probability distributions 37
•This process is known as standardization and gives the
position on a normal curve with μ=0 and σ=1, i.e., the
SND, Z.
•A Z-score is the number of standard deviations that a
given x value is above or below the mean.
38. Finding areas under the standard normal
distribution curve
The two steps are
Step 1 :
• Draw the normal distribution curve and shade the area.
Step 2:
Find the appropriate figure in the Procedure Table and follow the
directions given.
7/8/2023 Probability & probability distributions 38
40. Rules in computing probabilities
P[Z = a] = 0
P[Z ≤ a] obtained directly from the Z-table
P[Z ≥ a] = 1 – P[Z ≤ a]
P[Z ≥ -a] = P[Z ≤ +a]
P[Z ≤ -a] = P[Z ≥ +a]
P[a1 ≤ Z ≤ a2] = P[Z ≤ a2] – P[Z ≤ a1]
7/8/2023 Probability & probability distributions 40
41. Example 1
Find the area to the left of z = 2.06.
Solution
Step 1:
Draw the figure.
Step 2:
We are looking for the area under the standard normal
distribution to the left of z =2.06. look up the area in the table. It
is 0.9803. Hence, 98.03% of the area is less than z =2.06.
7/8/2023 Probability & probability distributions 41
42. Example 2
Find the area to the right of z = -1.19
Solution
Step 1: Draw the figure.
Step 2 : We are looking for the area to the right of z = -1.19. Look
up the area for z = -1.19. It is 0.1170. Subtract it from one. 1-
0.1170 = 0.8830. Hence, 88.30% of the area under the standard
normal distribution curve is to the right of z = -1.19.
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43. Continued
Example 3:
What is the probability that a z picked at random from the
population of z’s will have a value between -2.55 and +2.55 ?
Answer : P (-2.55 < z < +2.55)
= 0.9946 – 0.0054
= 0.9892
Example 4:
What proportion of z value are between -2.47 and 1.53?
Answer : P (-2.47 ≤ z ≤ 1.53)
= 0.9370 – 0.0068
= 0.9302
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44. Exercises
1. Find the area between z =1.68 and z =-1.37. Ans =
0.8682
2. Find the probability for each.
a) P(0 < z <2.32) = 0.4898
b) P(z < 1.65) = 0.9505
c) P(z >1.91)= 0.0281
3. Find two z values so that 48% of the middle area is bounded
by them?
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