Here are the key steps to solve this problem:
1) Draw the standard normal curve
2) The probability is the area between -2.55 and 2.55
3) From the standard normal table:
P(Z ≤ 2.55) = 0.9938
P(Z ≤ -2.55) = 0.0049
4) Use the area property:
P(-2.55 ≤ Z ≤ 2.55) = P(Z ≤ 2.55) - P(Z ≤ -2.55)
= 0.9938 - 0.0049
= 0.9889
Therefore, the probability that a z value will be between -2.55 and 2
In this slide, variables types, probability theory behind the algorithms and its uses including distribution is explained. Also theorems like bayes theorem is also explained.
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 ...
In this slide, variables types, probability theory behind the algorithms and its uses including distribution is explained. Also theorems like bayes theorem is also explained.
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 ...
Nursing education
And I will be home in a little after the meeting 🙂🙂🙂🙂🙂🙂 you want to do it is very important for you to comet ☄️ over and get it is very important to me some of your voice hi I will send you a picture of the money we have to buy you a picture of the money to buy a house in the morning and I can speak to buy you a new one for you to see the money we have to do it is very important to buy a house in the next few days are you working on the money we will be there around you and your family are doing well and I will send it to you tomorrow we will be home tomorrow we will be there is strategy for you to see the money 🤑 you want to do it is best to buy a house with you but I think it's the next few weeks are you doing today I will send you a link to the money we will have to do the next few weeks are you feeling today I think I will have to do it is physiology of you to see 🙈🙈 you are you working now or poikilocytosis the money 🤑🤑 you want me some of you have to do it is physiology lab 🧪 you are in the next few days are you working on the money we will have to get you a picture when I get it is physiology of the year you can you send me the link for Bsc Nursing education and your
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
1. Measurements of Health and disease
Probability & probability distributions
1/3/2023 Probability & probability distributions 1
Emiru Merdassa(MSc, Assistant professor)
2. 2
Definition
Probability: the chance that an uncertain event
will occur (always between 0 and 1)
0 ≤ P(A) ≤ 1 For any event A
Certain
Impossible
0.5
1
0
1/3/2023
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.
Probability & probability distributions
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
3
1/3/2023
(continued)
Probability & probability distributions
4. 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
(continued)
A B
AB
S
1/3/2023 Probability & probability distributions
5. 5
Important terms
• Two events A and B are mutually exclusive if they cannot both
happen at the same time
• P (A ∩ B) = 0
(continued)
A B
S
1/3/2023 Probability & probability distributions
6. 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
(continued)
A B
The entire shaded area
represents
A U B
S
1/3/2023 Probability & probability distributions
7. 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 ഥ
A
(continued)
A
S
A
1/3/2023 Probability & probability distributions
8. 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]
1/3/2023 Probability & probability distributions
9. 9
(continued)
Example
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 =
1/3/2023 Probability & probability distributions
10. 10
Mutually exclusive:
o A and B are not mutually exclusive
• The outcomes 4 and 6 are common to both
Collectively exhaustive:
o A and B are not collectively exhaustive
• A U B does not contain 1 or 3
(continued)
Example
S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]
1/3/2023 Probability & probability distributions
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:
O The outcomes on the first and second coin tosses are
independent
11
1/3/2023 Probability & probability distributions
12. Continued…
Two categories of probability
1. Objective probability and
2. Subjective probability
1) Objective probability
a) Classical probability &
b) Relative frequency probability.
12
1/3/2023 Probability & probability distributions
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
13
1/3/2023 Probability & probability distributions
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.
14
1/3/2023 Probability & probability distributions
15. Example
• 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.
15
1/3/2023 Probability & probability distributions
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%.
1/3/2023 16
(continued)
Probability & probability distributions
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.
17
1/3/2023 Probability & probability distributions
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)
1/3/2023 18
Probability & probability distributions
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.
1/3/2023 19
Probability & probability distributions
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)
20
1/3/2023 Probability & probability distributions
21. Example
The probabilities representing years of schooling
completed by mothers of newborn infants
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
1/3/2023 Probability & probability distributions
22. Continued
• 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
22
1/3/2023 Probability & probability distributions
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+
)?
1/3/2023 23
Probability & probability distributions
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.
25
1/3/2023 Probability & probability distributions
26. Normal distribution
1/3/2023 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
Probability & probability distributions
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.
1/3/2023 27
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
28
1/3/2023 Probability & probability distributions
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.
31
1/3/2023 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 .
32
1/3/2023 Probability & probability distributions
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.
34
1/3/2023 Probability & probability distributions
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
35
1/3/2023 Probability & probability distributions
36. Continued
– 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−
1/3/2023 36
Value x
50 65 80 95 110 125 140 155 171
-4 -3 -2 -1 0 1 2 3 4
Probability & probability distributions
37. Continued
– 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.
1/3/2023 Probability & probability distributions 37
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.
1/3/2023 38
Probability & probability distributions
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]
1/3/2023 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.
1/3/2023 41
Probability & probability distributions
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.
1/3/2023 42
Probability & probability distributions
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
1/3/2023 Probability & probability distributions 43
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?
1/3/2023 44
Probability & probability distributions