#TheProbabilityLifeSaver...
I am planning a picnic today, but the morning is cloudy. Oh no! 50% of all rainy days start off cloudy!
What is the probability/chance of rain during the day?
Shall I go for Picnic or not!
Also, I am too much crazy for fruit salad. "My fruit salad is a combination of apples, grapes and bananas" I don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", itās the same fruit salad. #Combination
We daily use probability concepts in our routine life. But we canāt think it is Statistics. just think little bit about statistics if we apply each & every statistical concepts in everyday life then what will happen...
#ApnaSapnaMoneyMoney #BreadButterHoney or Something more than this
#CuriosityRight
In this PPT you will see Probability & its importance
#RealLifeApplications Concept of events.
#Probability Rules #Events
#Conditional Probability
#Bayesā Theorem
#Permutation and Combination
#HowToCalculateProbability #DecisionMaking #PictorialView
#MakeFunWithProbExamples #Statistics #YogitaKolekar
#TheProbabilityLifeSaver...
I am planning a picnic today, but the morning is cloudy. Oh no! 50% of all rainy days start off cloudy!
What is the probability/chance of rain during the day?
Shall I go for Picnic or not!
Also, I am too much crazy for fruit salad. "My fruit salad is a combination of apples, grapes and bananas" I don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", itās the same fruit salad. #Combination
We daily use probability concepts in our routine life. But we canāt think it is Statistics. just think little bit about statistics if we apply each & every statistical concepts in everyday life then what will happen...
#ApnaSapnaMoneyMoney #BreadButterHoney or Something more than this
#CuriosityRight
In this PPT you will see Probability & its importance
#RealLifeApplications Concept of events.
#Probability Rules #Events
#Conditional Probability
#Bayesā Theorem
#Permutation and Combination
#HowToCalculateProbability #DecisionMaking #PictorialView
#MakeFunWithProbExamples #Statistics #YogitaKolekar
Probability - Question Bank for Class/Grade 10 maths.Let's Tute
Ā
Probability - Question Bank for Class/Grade 10 maths.
Watch videos on our youtube channel -
www.youtube.com/letstute.
And find related study material on our website -
www.letstute.com.
Here is the basic introduction to the probability used in my Analysis of Algorithms course at the Cinvestav Guadalajara. They go from the basic axioms to the Expected Value and Variance.
Some Basic concepts of Probability along with advanced concepts on Medical probability & Probability in Gambling. A lot of Sample Questions and Practice Questions will help you understand and apply the concepts in real life.
This presentation will clarify all your basic concepts of Probability. It includes Random Experiment, Sample Space, Event, Complementary event, Union - Intersection and difference of events, favorable cases, probability definitions, conditional probability, Bayes theorem
ļŗPlease Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 4: Probability
4.1: Basic Concepts of Probability
I am Josh U. I am a Probability Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from St. Edwardās University, USA.
I have been helping students with their homework for the past 5 years. I solve assignments related to Probability. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
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Probability - Question Bank for Class/Grade 10 maths.Let's Tute
Ā
Probability - Question Bank for Class/Grade 10 maths.
Watch videos on our youtube channel -
www.youtube.com/letstute.
And find related study material on our website -
www.letstute.com.
Here is the basic introduction to the probability used in my Analysis of Algorithms course at the Cinvestav Guadalajara. They go from the basic axioms to the Expected Value and Variance.
Some Basic concepts of Probability along with advanced concepts on Medical probability & Probability in Gambling. A lot of Sample Questions and Practice Questions will help you understand and apply the concepts in real life.
This presentation will clarify all your basic concepts of Probability. It includes Random Experiment, Sample Space, Event, Complementary event, Union - Intersection and difference of events, favorable cases, probability definitions, conditional probability, Bayes theorem
ļŗPlease Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 4: Probability
4.1: Basic Concepts of Probability
I am Josh U. I am a Probability Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from St. Edwardās University, USA.
I have been helping students with their homework for the past 5 years. I solve assignments related to Probability. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Probability Assignments.
This powerpoint was used in my 7th and 8th grade classes to review the fundamental counting principle used in our probability unit. There are three independent practice problems at the end.
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About Probability.. Here you Accure about Probability. It describes about toss, card,and many others example to teach probability clearly and helps to understand a clear concept. This students can easily get knowledge from the presentation slide. And it is made from the basic. It also discuss about some rules of probability
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Elementary Statistics Practice Test 2 Solutions
Chapter 4: Probability
I am Christopher, T.I am a Mathematics Assignment Expert at eduassignmenthelp.com. I hold a PhD. in Mathematics, University of Alberta, Canada. I have been helping students with their Assignments for the past 7 years. I solve assignments related to Mathematics.
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Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
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In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
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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.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
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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?
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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2. A gambling experiment
ļ® Everyone in the room takes 2 cards
from the deck (keep face down)
ļ® Rules, most to least valuable:
ļ®
Pair of the same color (both red or both black)
ļ®
Mixed-color pair (1 red, 1 black)
ļ®
Any two cards of the same suit
ļ®
Any two cards of the same color
In the event of a tie, highest card wins (ace is top)
3. What do you want to bet?
ļ® Look at your two cards.
ļ® Will you fold or bet?
ļ® What is the most rational strategy given
your hand?
4. Rational strategy
ļ® There are N people in the room
ļ® What are the chances that someone in
the room has a better hand than you?
ļ® Need to know the probabilities of
different scenarios
ļ® Weāll return to this later in the lectureā¦
5. Probability
ļ® Probability ā the chance that an uncertain
event will occur (always between 0 and 1)
Symbols:
P(event A) = āthe probability that event A will occurā
P(red card) = āthe probability of a red cardā
P(~event A) = āthe probability of NOT getting event Aā [complement]
P(~red card) = āthe probability of NOT getting a red cardā
P(A & B) = āthe probability that both A and B happenā [joint probability]
P(red card & ace) = āthe probability of getting a red aceā
6. Assessing Probability
1. Theoretical/Classical probabilityābased on theory
(a priori understanding of a phenomena)
e.g.: theoretical probability of rolling a 2 on a standard die is 1/6
theoretical probability of choosing an ace from a standard deck
is 4/52
theoretical probability of getting heads on a regular coin is 1/2
2. Empirical probabilityābased on empirical data
e.g.: you toss an irregular die (probabilities unknown) 100 times and
find that you get a 2 twenty-five times; empirical probability of
rolling a 2 is 1/4
empirical probability of an Earthquake in Bay Area by 2032 is .
62 (based on historical data)
empirical probability of a lifetime smoker developing lung cancer
is 15 percent (based on empirical data)
7. Recent headlines on earthquake
probabiilitesā¦
http://www.guardian.co.uk/world/2011/may/26/italy-quake-expe
8. Computing theoretical
probabilities:counting methods
Great for gambling! Fun to compute!
If outcomes are equally likely to occurā¦
outcomesof#total
occurcanAwaysof#
)( =AP
Note: these are called ācounting methodsā because we have
to count the number of ways A can occur and the number
of total possible outcomes.
9. Counting methods: Example 1
0769.
52
4
deckin thecardsof#
deckin theacesof#
)aceandraw( ===P
Example 1: You draw one card from a deck of
cards. Whatās the probability that you draw an ace?
10. Counting methods: Example 2
Example 2. Whatās the probability that you draw 2 aces when you draw
two cards from the deck?
52
4
deckin thecardsof#
deckin theacesof#
)drawfirstonacedraw( ==P
51
3
deckin thecardsof#
deckin theacesof#
)toodrawsecondonaceandraw( ==P
51
3
x
52
4
ace)ANDacedraw( =ā“P
This is a ājoint probabilityāāweāll get back to this on Wednesday
13. Summary of Counting Methods
Counting methods for computing probabilities
With replacement
Without replacement
Permutationsā
order matters!
Combinationsā
Order doesnāt
matter
Without replacement
14. Summary of Counting Methods
Counting methods for computing probabilities
With replacement
Without replacement
Permutationsā
order matters!
15. PermutationsāOrder matters!
A permutation is an ordered arrangement of objects.
With replacement=once an event occurs, it can occur again
(after you roll a 6, you can roll a 6 again on the same die).
Without replacement=an event cannot repeat (after you draw
an ace of spades out of a deck, there is 0 probability of
getting it again).
16. Summary of Counting Methods
Counting methods for computing probabilities
With replacement
Permutationsā
order matters!
17. With Replacement ā Think coin tosses, dice, and DNA.
āmemorylessā ā After you get heads, you have an equally likely chance of getting a
heads on the next toss (unlike in cards example, where you canāt draw the same card
twice from a single deck).
Whatās the probability of getting two heads in a row (āHHā) when tossing a coin?
H
H
T
T
H
T
Toss 1:
2 outcomes
Toss 2:
2 outcomes 22
total possible outcomes: {HH, HT, TH, TT}
Permutationsāwith replacement
outcomespossible2
HHgetway to1
)( 2
=HHP
18. Whatās the probability of 3 heads in a row?
outcomespossible82
1
)( 3
=
=HHHP
Permutationsāwith replacement
H
H
T
T
H
T
Toss 1:
2 outcomes
Toss 2:
2 outcomes
Toss 3:
2 outcomes
H
T
H
T
H
T
H
T
HH
H
HHT
HTH
HTT
THH
THT
TTH
TTT
19. 36
1
6
66,rollway to1
)6,6( 2
=P
When you roll a pair of dice (or 1 die twice),
whatās the probability of rolling 2 sixes?
Whatās the probability of rolling a 5 and a 6?
36
2
6
6,5or5,6:ways2
)6&5( 2
==P
Permutationsāwith replacement
20. Summary: order matters, with
replacement
Formally, āorder mattersā and āwith
replacementāļ use powersļ
reventsof#the
nevent)peroutcomespossible(# =
21. Summary of Counting Methods
Counting methods for computing probabilities
Without replacement
Permutationsā
order matters!
22. Permutationsāwithout
replacement
Without replacementāThink cards (w/o reshuffling)
and seating arrangements.
Example: You are moderating a debate of
gubernatorial candidates. How many different ways
can you seat the panelists in a row? Call them
Arianna, Buster, Camejo, Donald, and Eve.
23. Permutationāwithout
replacement
ļ āTrial and errorā method:
Systematically write out all combinations:
A B C D E
A B C E D
A B D C E
A B D E C
A B E C D
A B E D C
.
.
.
Quickly becomes a pain!
Easier to figure out patterns using a the
probability tree!
25. Permutationāwithout
replacement
What if you had to arrange 5 people in only 3 chairs
(meaning 2 are out)?
==
!2
!5
12
12345
x
xxxx
E
B
A
C
D
E
A
B
D
A
B
C
D
Seat One:
5 possible
Seat Two:
Only 4 possible
E
B
D
Seat Three:
only 3 possible
)!35(
!5
ā
=345 xx
28. Summary: order matters,
without replacement
Formally, āorder mattersā and āwithout
replacementāļ use factorialsļ
)1)...(2)(1(or
)!(
!
draws)!orchairscardsorpeople(
cards)!orpeople(
+āāā
ā
=
ā
rnnnn
rn
n
rn
n
29. Practice problems:
1. A wine taster claims that she can distinguish
four vintages or a particular Cabernet. What
is the probability that she can do this by
merely guessing (she is confronted with 4
unlabeled glasses)? (hint: without
replacement)
2. In some states, license plates have six
characters: three letters followed by three
numbers. How many distinct such plates are
possible? (hint: with replacement)
30. Answer 1
1. A wine taster claims that she can distinguish four vintages or a particular
Cabernet. What is the probability that she can do this by merely
guessing (she is confronted with 4 unlabeled glasses)? (hint: without
replacement)
P(success) = 1 (thereās only way to get it right!) / total # of guesses she could make
Total # of guesses one could make randomly:
glass one: glass two: glass three: glass four:
4 choices 3 vintages left 2 left no ādegrees of freedomā left
ā“P(success) = 1 / 4! = 1/24 = .04167
= 4 x 3 x 2 x 1 = 4!
31. Answer 2
2. In some states, license plates have six characters: three letters
followed by three numbers. How many distinct such plates are
possible? (hint: with replacement)
263
different ways to choose the letters and 103
different ways to
choose the digitsĀ
ā“total number = 263
x 103
= 17,576 x 1000 = 17,576,000
32. Summary of Counting Methods
Counting methods for computing probabilities
Combinationsā
Order doesnāt
matter
Without replacement
34. Combinations
2)!252(
!52
2
5152
ā
=
x
How many two-card hands can I draw from a deck when order
does not matter (e.g., ace of spades followed by ten of clubs is
the same as ten of clubs followed by ace of spades)
.
.
.
Ā
Ā 52Ā cards 51Ā cards
.
.
.
Ā
35. Combinations
?
4849505152 xxxx
How many five-card hands can I draw from a deck when order
does not matter?
.
.
.
Ā
Ā
52Ā cards
51Ā cards
.
.
.
Ā
.
.
.
Ā
.
.
.
Ā
.
.
.
Ā
50Ā cards
49Ā cards
48Ā cards
39. Combinations
ļ® How many unique 2-card sets out of 52
cards?
ļ® 5-card sets?
ļ® r-card sets?
ļ® r-card sets out of n-cards?
!2)!252(
!52
2
5152
ā
=
x
!5)!552(
!52
!5
4849505152
ā
=
xxxx
!)!52(
!52
rrā
!)!(
!
rrn
nn
r ā
=ļ£·
ļ£ø
ļ£¶
ļ£¬
ļ£
ļ£«
40. Summary: combinations
If rĀ objectsĀ areĀ takenĀ fromĀ aĀ setĀ ofĀ n objectsĀ withoutĀ replacementĀ andĀ
disregardingĀ order,Ā howĀ manyĀ differentĀ samplesĀ areĀ possible?Ā
Formally, āorder doesnāt matterā and āwithout replacementāļ
use choosingļ
Ā
!)!(
!
rrn
nn
r ā
=ļ£·
ļ£ø
ļ£¶
ļ£¬
ļ£
ļ£«
41. ExamplesāCombinations
A lottery works by picking 6 numbers from 1 to 49. How
many combinations of 6 numbers could you choose?
816,983,13
!6!43
!4949
6
==ļ£·
ļ£ø
ļ£¶
ļ£¬
ļ£
ļ£«
Which of course means that your probability of winning is 1/13,983,816!
43. Summary of Counting
Methods
Counting methods for computing probabilities
With replacement: nr
Permutationsā
order matters!
Without replacement:
n(n-1)(n-2)ā¦(n-r+1)=
Combinationsā
Order doesnāt
matter
Without
replacement:
)!(
!
rn
n
ā
!)!(
!
rrn
nn
r ā
=ļ£·
ļ£ø
ļ£¶
ļ£¬
ļ£
ļ£«
44. Gambling, revisited
ļ® What are the probabilities of the
following hands?
ļ®
Pair of the same color
ļ®
Pair of different colors
ļ®
Any two cards of the same suit
ļ®
Any two cards of the same color
45. Pair of the same color?
ļ® P(pair of the same color) =
nscombinatioĀ cardĀ twoofĀ #Ā total
colorĀ sameĀ ofĀ pairsĀ #
ā
Numerator = red aces, black aces; red kings, black kings;
etc.ā¦= 2x13 = 26
1326
2
52x51
rDenominato 252 === C
chanceĀ 1.96%Ā Ā
1326
26
Ā Ā color)Ā sameĀ Ā theofP(pairĀ Ā So, ==
46. Any old pair?
ļ® P(any pair) =
1326nscombinatiocardtwoof#total
pairs#
=ā
chance5.9%
1326
78
pair)P(any ==ā“
pairspossibletotal7813x6
...
6
2
34
!2!2
4!
Ckingsofpairspossibledifferentofnumber
6
2
34
!2!2
4!
Cacesofpairspossibledifferentofnumber
24
24
=
====
====
x
x
47. Two cards of same suit?
3124784
11!2!
13!
suits4C:Numerator 213 === xxx
chance23.5%
1326
312
suit)sametheofcardsP(two ==ā“
48. Two cards of same color?
Numerator: 26C2 x 2 colors = 26!/(24!2!) = 325 x 2 = 650
Denominator = 1326
So, P (two cards of the same color) = 650/1326 = 49% chance
A little non-intuitive? Hereās another way to look at itā¦
.
.
.
Ā
Ā 52Ā cards
26Ā redĀ branches
26Ā blackĀ branches
FromĀ aĀ RedĀ branch:Ā 26Ā blackĀ left,Ā 25Ā redĀ left
.
.
.
Ā
FromĀ aĀ BlackĀ branch:Ā 26Ā redĀ left,Ā 25Ā blackĀ left
26x25 RR
26x26 RB
26x26 BR
26x25 BB
50/102
Not
quite
50/100
49. Rational strategy?
ļ® To bet or fold?
ļ® It would be really complicated to take into
account the dependence between hands in the
class (since we all drew from the same deck), so
weāre going to fudge this and pretend that
everyone had equal probabilities of each type of
hand (pretend we have āindependenceā)ā¦Ā
ļ® Just to get a rough idea...
50. Rational strategy?
**Trick! P(at least 1) = 1- P(0)
P(at least one same-color pair in the class)=
1-P(no same-color pairs in the whole class)=
paircolor-sameoneleastatofchance.4%55.446-1(.98)-1 40
==
40
)98(.)....98(.*)98(.*)98(.class)wholein thepairscolor-sameP(no
.98.0196-1pair)color-sameagettdon'P(I
==
ā =
51. Rational strategy?
P(at least one pair)= 1-P(no pairs)=
1-(.94)40
=1-8%=92% chance
P(>=1 same suit)= 1-P(all different suits)=
1-(.765)40
=1-.00002 ~ 100%
P(>=1 same color) = 1-P(all different colors)=
1-(.51) 40
=1-.000000000002 ~ 100%
52. Rational strategyā¦
ļ® Fold unless you have a same-color pair or a
numerically high pair (e.g., Queen, King,
Ace).
How does this compare to class?
-anyone with a same-color pair?
-any pair?
-same suit?
-same color?
53. Practice problem:
ļ® A classic problem: āThe Birthday Problem.ā Whatās
the probability that two people in a class of 25 have
the same birthday? (disregard leap years)
What would you guess is the probability?
54. Birthday Problem Answer
1. A classic problem: āThe Birthday Problem.ā Whatās the
probability that two people in a class of 25 have the same
birthday? (disregard leap years)
Ā **Trick! 1- P(none) = P(at least one)
Use complement to calculate answer. Itās easier to calculate 1- P(no
matches) = the probability that at least one pair of people have the
same birthday.
Whatās the probability of no matches?
Denominator: how many sets of 25 birthdays are there?
--with replacement (order matters)
36525
Numerator: how many different ways can you distribute 365 birthdays
to 25 people without replacement?
--order matters, without replacement:
[365!/(365-25)!]= [365 x 364 x 363 x 364 x ā¦.. (365-24)]
Ā ā“ P(no matches) = [365 x 364 x 363 x 364 x ā¦.. (365-24)] / 36525
55. Use SAS as a calculator
Ā Use SAS as calculatorā¦ (my calculator wonāt do factorials as high as 365, so I had to
improvise by using a loopā¦which youāll learn later in HRP 223):
Ā
%LET num = 25; *set number in the class;
data null;
top=1; *initialize numerator;
do j=0 to (&num-1) by 1;
top=(365-j)*top;
end;
BDayProb=1-(top/365**&num);
put BDayProb;
run;
Ā From SAS log:
Ā
0.568699704, so 57% chance!
56. For class of 40 (our class)?
For class of 40?
10 %LET num = 40; *set number in the class;
11 data null;
12 top=1; *initialize numerator;
13 do j=0 to (&num-1) by 1;
14 top=(365-j)*top;
15 end;
16 BDayProb=1-(top/365**&num);
17 put BDayProb;
18 run;
0.891231809, i.e. 89% chance of a
match!