The document describes the game of craps and various bets that can be made. It provides the rules and probabilities associated with different outcomes. For a standard craps bet that pays even money, the probability of winning is 5/9 and losing is 4/9. Simulation of 1,000 $1 bets results in an expected net loss, with actual results varying randomly based on dice rolls. Bets with higher payouts have lower probabilities of winning to offset the house advantage.
An investigation of the mathematics of casino gaming particularly how quantities like house advantage, expected value, win, hold, drop, and hold percentage are used by casinos.
An investigation of the mathematics of casino gaming particularly how quantities like house advantage, expected value, win, hold, drop, and hold percentage are used by casinos.
This is a beginners guide to the game, and even some great links for free money, real, to start creating your own bank. This presentation was made from the portuguese one created to my own blog.
Thank you for reading
Márcio Guerra
10 centuries of the history of the poker. Basic rules and mathematical expectation. What should you do to win more often.
For information on other games visit https://nz-casinos.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.
This is a beginners guide to the game, and even some great links for free money, real, to start creating your own bank. This presentation was made from the portuguese one created to my own blog.
Thank you for reading
Márcio Guerra
10 centuries of the history of the poker. Basic rules and mathematical expectation. What should you do to win more often.
For information on other games visit https://nz-casinos.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.
Can you teach coding to kids in a mobile game app in local languages. Do you need to be good in English to learn coding in R or Python?
How young can we train people in coding-
something we worked on for six months but now we are giving up due to lack of funds is this idea.
Feel free to use it, it is licensed cc-by-sa
Chatty Kathy - UNC Bootcamp Final Project Presentation - Final Version - 5.23...John Andrews
SlideShare Description for "Chatty Kathy - UNC Bootcamp Final Project Presentation"
Title: Chatty Kathy: Enhancing Physical Activity Among Older Adults
Description:
Discover how Chatty Kathy, an innovative project developed at the UNC Bootcamp, aims to tackle the challenge of low physical activity among older adults. Our AI-driven solution uses peer interaction to boost and sustain exercise levels, significantly improving health outcomes. This presentation covers our problem statement, the rationale behind Chatty Kathy, synthetic data and persona creation, model performance metrics, a visual demonstration of the project, and potential future developments. Join us for an insightful Q&A session to explore the potential of this groundbreaking project.
Project Team: Jay Requarth, Jana Avery, John Andrews, Dr. Dick Davis II, Nee Buntoum, Nam Yeongjin & Mat Nicholas
As Europe's leading economic powerhouse and the fourth-largest hashtag#economy globally, Germany stands at the forefront of innovation and industrial might. Renowned for its precision engineering and high-tech sectors, Germany's economic structure is heavily supported by a robust service industry, accounting for approximately 68% of its GDP. This economic clout and strategic geopolitical stance position Germany as a focal point in the global cyber threat landscape.
In the face of escalating global tensions, particularly those emanating from geopolitical disputes with nations like hashtag#Russia and hashtag#China, hashtag#Germany has witnessed a significant uptick in targeted cyber operations. Our analysis indicates a marked increase in hashtag#cyberattack sophistication aimed at critical infrastructure and key industrial sectors. These attacks range from ransomware campaigns to hashtag#AdvancedPersistentThreats (hashtag#APTs), threatening national security and business integrity.
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Our comprehensive report delves into these challenges, using a blend of open-source and proprietary data collection techniques. By monitoring activity on critical networks and analyzing attack patterns, our team provides a detailed overview of the threats facing German entities.
This report aims to equip stakeholders across public and private sectors with the knowledge to enhance their defensive strategies, reduce exposure to cyber risks, and reinforce Germany's resilience against cyber threats.
Techniques to optimize the pagerank algorithm usually fall in two categories. One is to try reducing the work per iteration, and the other is to try reducing the number of iterations. These goals are often at odds with one another. Skipping computation on vertices which have already converged has the potential to save iteration time. Skipping in-identical vertices, with the same in-links, helps reduce duplicate computations and thus could help reduce iteration time. Road networks often have chains which can be short-circuited before pagerank computation to improve performance. Final ranks of chain nodes can be easily calculated. This could reduce both the iteration time, and the number of iterations. If a graph has no dangling nodes, pagerank of each strongly connected component can be computed in topological order. This could help reduce the iteration time, no. of iterations, and also enable multi-iteration concurrency in pagerank computation. The combination of all of the above methods is the STICD algorithm. [sticd] For dynamic graphs, unchanged components whose ranks are unaffected can be skipped altogether.
Explore our comprehensive data analysis project presentation on predicting product ad campaign performance. Learn how data-driven insights can optimize your marketing strategies and enhance campaign effectiveness. Perfect for professionals and students looking to understand the power of data analysis in advertising. for more details visit: https://bostoninstituteofanalytics.org/data-science-and-artificial-intelligence/
Opendatabay - Open Data Marketplace.pptxOpendatabay
Opendatabay.com unlocks the power of data for everyone. Open Data Marketplace fosters a collaborative hub for data enthusiasts to explore, share, and contribute to a vast collection of datasets.
First ever open hub for data enthusiasts to collaborate and innovate. A platform to explore, share, and contribute to a vast collection of datasets. Through robust quality control and innovative technologies like blockchain verification, opendatabay ensures the authenticity and reliability of datasets, empowering users to make data-driven decisions with confidence. Leverage cutting-edge AI technologies to enhance the data exploration, analysis, and discovery experience.
From intelligent search and recommendations to automated data productisation and quotation, Opendatabay AI-driven features streamline the data workflow. Finding the data you need shouldn't be a complex. Opendatabay simplifies the data acquisition process with an intuitive interface and robust search tools. Effortlessly explore, discover, and access the data you need, allowing you to focus on extracting valuable insights. Opendatabay breaks new ground with a dedicated, AI-generated, synthetic datasets.
Leverage these privacy-preserving datasets for training and testing AI models without compromising sensitive information. Opendatabay prioritizes transparency by providing detailed metadata, provenance information, and usage guidelines for each dataset, ensuring users have a comprehensive understanding of the data they're working with. By leveraging a powerful combination of distributed ledger technology and rigorous third-party audits Opendatabay ensures the authenticity and reliability of every dataset. Security is at the core of Opendatabay. Marketplace implements stringent security measures, including encryption, access controls, and regular vulnerability assessments, to safeguard your data and protect your privacy.
Adjusting primitives for graph : SHORT REPORT / NOTESSubhajit Sahu
Graph algorithms, like PageRank Compressed Sparse Row (CSR) is an adjacency-list based graph representation that is
Multiply with different modes (map)
1. Performance of sequential execution based vs OpenMP based vector multiply.
2. Comparing various launch configs for CUDA based vector multiply.
Sum with different storage types (reduce)
1. Performance of vector element sum using float vs bfloat16 as the storage type.
Sum with different modes (reduce)
1. Performance of sequential execution based vs OpenMP based vector element sum.
2. Performance of memcpy vs in-place based CUDA based vector element sum.
3. Comparing various launch configs for CUDA based vector element sum (memcpy).
4. Comparing various launch configs for CUDA based vector element sum (in-place).
Sum with in-place strategies of CUDA mode (reduce)
1. Comparing various launch configs for CUDA based vector element sum (in-place).
Data Centers - Striving Within A Narrow Range - Research Report - MCG - May 2...pchutichetpong
M Capital Group (“MCG”) expects to see demand and the changing evolution of supply, facilitated through institutional investment rotation out of offices and into work from home (“WFH”), while the ever-expanding need for data storage as global internet usage expands, with experts predicting 5.3 billion users by 2023. These market factors will be underpinned by technological changes, such as progressing cloud services and edge sites, allowing the industry to see strong expected annual growth of 13% over the next 4 years.
Whilst competitive headwinds remain, represented through the recent second bankruptcy filing of Sungard, which blames “COVID-19 and other macroeconomic trends including delayed customer spending decisions, insourcing and reductions in IT spending, energy inflation and reduction in demand for certain services”, the industry has seen key adjustments, where MCG believes that engineering cost management and technological innovation will be paramount to success.
MCG reports that the more favorable market conditions expected over the next few years, helped by the winding down of pandemic restrictions and a hybrid working environment will be driving market momentum forward. The continuous injection of capital by alternative investment firms, as well as the growing infrastructural investment from cloud service providers and social media companies, whose revenues are expected to grow over 3.6x larger by value in 2026, will likely help propel center provision and innovation. These factors paint a promising picture for the industry players that offset rising input costs and adapt to new technologies.
According to M Capital Group: “Specifically, the long-term cost-saving opportunities available from the rise of remote managing will likely aid value growth for the industry. Through margin optimization and further availability of capital for reinvestment, strong players will maintain their competitive foothold, while weaker players exit the market to balance supply and demand.”
Data Centers - Striving Within A Narrow Range - Research Report - MCG - May 2...
Craps
1. 2/12/2016 Craps
http://www.math.uah.edu/stat/games/Craps.html 1/7
Random > 12. Games of Chance > 1 2 3 4 5 6 7 8 9 10 11
4. Craps
The Basic Game
Craps is a popular casino game, because of its complexity and because of the rich variety of bets that can be made.
A typical craps table
According to Richard Epstein, craps is descended from an earlier game known as Hazard, that dates to the Middle Ages.
The formal rules for Hazard were established by Montmort early in the 1700s. The origin of the name craps is shrouded in
doubt, but it may have come from the English crabs or from the French Crapeaud (for toad).
From a mathematical point of view, craps is interesting because it is an example of a random experiment that takes place
in stages; the evolution of the game depends critically on the outcome of the first roll. In particular, the number of rolls is a
random variable.
Definitions
The rules for craps are as follows:
1. The player (known as the shooter) rolls a pair of fair dice
a. If the sum is 7 or 11 on the first throw, the shooter wins; this event is called a natural.
b. If the sum is 2, 3, or 12 on the first throw, the shooter loses; this event is called craps.
c. If the sum is 4, 5, 6, 8, 9, or 10 on the first throw, this number becomes the shooter's point. The shooter continues
rolling the dice until either she rolls the point again (in which case she wins) or rolls a 7 (in which case she loses).
As long as the shooter wins, or loses by rolling craps, she retrains the dice and continues. Once she loses by failing to
make her point, the dice are passed to the next shooter.
2. 2/12/2016 Craps
http://www.math.uah.edu/stat/games/Craps.html 2/7
Let us consider the game of craps mathematically. Our basic assumption, of course, is that the dice are fair and that the
outcomes of the various rolls are independent. Let denote the (random) number of rolls in the game and let
denote the outcome of the th roll for . Finally, let , the sum of the scores on the th
roll, and let denote the indicator variable that the shooter wins.
2. In the craps experiment, press single step a few times and observe the outcomes. Make sure that you understand the
rules of the game.
The Probability of Winning
We will compute the probability that the shooter wins in stages, based on the outcome of the first roll.
3. The sum of the scores on a given roll has the probability density function in the following table:
2 3 4 5 6 7 8 9 10 11 12
The probability that the player makes her point can be computed using a simple conditioning argument. For example,
suppose that the player throws 4 initially, so that 4 is the point. The player continues until she either throws 4 again or
throws 7. Thus, the final roll will be an element of the following set:
Since the dice are fair, these outcomes are equally likely, so the probability that the player makes her 4 point is . A
similar argument can be used for the other points. Here are the results:
4. The probabilities of making the point are given in the following table:
4 5 6 8 9 10
5. The probability that the shooter wins is
Proof:
Note that craps is nearly a fair game.
Bets
There is a bewildering variety of bets that can be made in craps. In the exercises in this subsection, we will discuss some
typical bets and compute the probability density function, mean, and standard deviation of each. (Most of these bets are
illustrated in the picture of the craps table above). Note however, that some of the details of the bets and, in particular the
payout odds, vary from one casino to another. Of course the expected value of any bet is inevitably negative (for the
gambler), and thus the gambler is doomed to lose money in the long run. Nonetheless, as we will see, some bets are better
than others.
Pass and Don't Pass
A pass bet is a bet that the shooter will win and pays .
6. Let denote the winnings from a unit pass bet. Then
N ( , )Xi Yi
i i ∈ {1, 2, … , N} = +Zi Xi Yi i
I
Z
z
P(Z = z)
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
= {(1, 3), (2, 2), (3, 1), (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}S4
3
9
z
z
P(I = 1 ∣ = z)Z1
3
9
4
10
5
11
5
11
4
10
3
9
P(I = 1) = ≈ 0.49292
244
495
1 : 1
W
P
251
P
244
3. 2/12/2016 Craps
http://www.math.uah.edu/stat/games/Craps.html 3/7
a. ,
b.
c.
7. In the craps experiment, select the pass bet. Run the simulation 1000 times and compare the empirical density
function and moments of to the true probability density function and moments. Suppose that you bet $1 on each of
the 1000 games. What would your net winnings be?
A don't pass bet is a bet that the shooter will lose, except that 12 on the first throw is excluded (that is, the shooter loses, of
course, but the don't pass better neither wins nor loses). This is the meaning of the phrase don't pass bar double 6 on the
craps table. The don't pass bet also pays .
8. Let denote the winnings for a unit don't pass bet. Then
a. , ,
b.
c.
Thus, the don't pass bet is slightly better for the gambler than the pass bet.
9. In the craps experiment, select the don't pass bet. Run the simulation 1000 times and compare the empirical density
function and moments of to the true probability density function and moments. Suppose that you bet $1 on each of
the 1000 games. What would your net winnings be?
The come bet and the don't come bet are analogous to the pass and don't pass bets, respectively, except that they are made
after the point has been established.
Field
A field bet is a bet on the outcome of the next throw. It pays if 3, 4, 9, 10, or 11 is thrown, if 2 or 12 is thrown,
and loses otherwise.
10. Let denote the winnings for a unit field bet. Then
a. , ,
b.
c.
11. In the craps experiment, select the field bet. Run the simulation 1000 times and compare the empirical density
function and moments of to the true probability density function and moments. Suppose that you bet $1 on each of
the 1000 games. What would your net winnings be?
Seven and Eleven
A 7 bet is a bet on the outcome of the next throw. It pays if a 7 is thrown. Similarly, an 11 bet is a bet on the outcome
of the next throw, and pays if an 11 is thrown. In spite of the romance of the number 7, the next exercise shows that
the 7 bet is one of the worst bets you can make.
P(W = −1) =
251
495
P(W = 1) =
244
495
E(W ) = − ≈ −0.0141
7
495
sd(W ) ≈ 0.9999
W
1 : 1
W
P(W = −1) =
244
495
P(W = 0) =
1
36
P(W = 1) =
949
1980
E(W ) = − ≈ −0.01363
27
1980
sd(W ) ≈ 0.9859
W
1 : 1 2 : 1
W
P(W = −1) =
5
9
P(W = 1) =
7
18
P(W = 2) =
1
18
E(W ) = − ≈ −0.0556
1
18
sd(W ) ≈ 1.0787
W
4 : 1
15 : 1
4. 2/12/2016 Craps
http://www.math.uah.edu/stat/games/Craps.html 4/7
12. Let denote the winnings for a unit 7 bet. Then
a. ,
b.
c.
13. In the craps experiment, select the 7 bet. Run the simulation 1000 times and compare the empirical density function
and moments of to the true probability density function and moments. Suppose that you bet $1 on each of the 1000
games. What would your net winnings be?
14. Let denote the winnings for a unit 11 bet. Then
a. ,
b.
c.
15. In the craps experiment, select the 11 bet. Run the simulation 1000 times and compare the empirical density
function and moments of to the true probability density function and moments. Suppose that you bet $1 on each of
the 1000 games. What would your net winnings be?
Craps
All craps bets are bets on the next throw. The basic craps bet pays if 2, 3, or 12 is thrown. The craps 2 bet pays
if a 2 is thrown. Similarly, the craps 12 bet pays if a 12 is thrown. Finally, the craps 3 bet pays if a 3
is thrown.
16. Let denote the winnings for a unit craps bet. Then
a. ,
b.
c.
17. In the craps experiment, select the craps bet. Run the simulation 1000 times and compare the empirical density
function and moments of to the true probability density function and moments. Suppose that you bet $1 on each of
the 1000 games. What would your net winnings be?
18. Let denote the winnings for a unit craps 2 bet or a unit craps 12 bet. Then
a. ,
b.
c.
19. In the craps experiment, select the craps 2 bet. Run the simulation 1000 times and compare the empirical density
function and moments of to the true probability density function and moments. Suppose that you bet $1 on each of
the 1000 games. What would your net winnings be?
W
P(W = −1) =
5
6
P(W = 4) =
1
6
E(W ) = − ≈ −0.1667
1
6
sd(W ) ≈ 1.8634
W
W
P(W = −1) =
17
18
P(W = 15) =
1
18
E(W ) = − ≈ −0.1111
1
9
sd(W ) ≈ 3.6650
W
7 : 1
30 : 1 30 : 1 15 : 1
W
P(W = −1) =
8
9
P(W = 7) =
1
9
E(W ) = − ≈ −0.1111
1
9
sd(W ) ≈ 5.0944
W
W
P(W = −1) =
35
36
P(W = 30) =
1
36
E(W ) = − ≈ −0.1389
5
36
sd(W ) = 5.0944
W
5. 2/12/2016 Craps
http://www.math.uah.edu/stat/games/Craps.html 5/7
20. In the craps experiment, select the craps 12 bet. Run the simulation 1000 times and compare the empirical density
function and moments of to the true probability density function and moments. Suppose that you bet $1 on each of
the 1000 games. What would your net winnings be?
21. Let denote the winnings for a unit craps 3 bet. Then
a. ,
b.
c.
22. In the craps experiment, select the craps 3 bet. Run the simulation 1000 times and compare the empirical density
function and moments of to the true probability density function and moments. Suppose that you bet $1 on each of
the 1000 games. What would your net winnings be?
Thus, of the craps bets, the basic craps bet and the craps 3 bet are best for the gambler, and the craps 2 and craps 12 are the
worst.
Big Six and Big Eight
The big 6 bet is a bet that 6 is thrown before 7. Similarly, the big 8 bet is a bet that 8 is thrown before 7. Both pay even
money .
23. Let denote the winnings for a unit big 6 bet or a unit big 8 bet. Then
a. ,
b.
c.
24. In the craps experiment, select the big 6 bet. Run the simulation 1000 times and compare the empirical density
function and moments of to the true probability density function and moments. Suppose that you bet $1 on each of
the 1000 games. What would your net winnings be?
25. In the craps experiment, select the big 8 bet. Run the simulation 1000 times and compare the empirical density
function and moments of to the true probability density function and moments. Suppose that you bet $1 on each of
the 1000 games. What would your net winnings be?
Hardway Bets
A hardway bet can be made on any of the numbers 4, 6, 8, or 10. It is a bet that the chosen number will be thrown “the
hardway” as , before 7 is thrown and before the chosen number is thrown in any other combination. Hardway
bets on 4 and 10 pay , while hardway bets on 6 and 8 pay .
26. Let denote the winnings for a unit hardway 4 or hardway 10 bet. Then
a. ,
b.
c.
W
W
P(W = −1) =
17
18
P(W = 15) =
1
18
E(W ) = − ≈ −0.1111
1
9
sd(W ) ≈ 3.6650
W
1 : 1
W
P(W = −1) =
6
11
P(W = 1) =
5
11
E(W ) = − ≈ −0.0909
1
11
sd(W ) ≈ 0.9959
W
W
n
(n/2, n/2)
7 : 1 9 : 1
W
P(W = −1) =
8
9
P(W = 7) =
1
9
E(W ) = − ≈ −0.1111
1
9
sd(W ) = 2.5142
6. 2/12/2016 Craps
http://www.math.uah.edu/stat/games/Craps.html 6/7
27. In the craps experiment, select the hardway 4 bet. Run the simulation 1000 times and compare the empirical density
function and moments of to the true probability density function and moments. Suppose that you bet $1 on each of
the 1000 games. What would your net winnings be?
28. In the craps experiment, select the hardway 10 bet. Run the simulation 1000 times and compare the empirical
density function and moments of to the true probability density function and moments. Suppose that you bet $1 on
each of the 1000 games. What would your net winnings be?
29. Let denote the winnings for a unit hardway 6 or hardway 8 bet. Then
a. ,
b.
c.
30. In the craps experiment, select the hardway 6 bet. Run the simulation 1000 times and compare the empirical density
and moments of to the true density and moments. Suppose that you bet $1 on each of the 1000 games. What would
your net winnings be?
31. In the craps experiment, select the hardway 8 bet. Run the simulation 1000 times and compare the empirical density
function and moments of to the true probability density function and moments. Suppose that you bet $1 on each of
the 1000 games. What would your net winnings be?
Thus, the hardway 6 and 8 bets are better than the hardway 4 and 10 bets for the gambler, in terms of expected value.
The Distribution of the Number of Rolls
Next let us compute the distribution and moments of the number of rolls in a game of craps. This random variable is of
no special interest to the casino or the players, but provides a good mathematically exercise. By definition, if the shooter
wins or loses on the first roll, . Otherwise, the shooter continues until she either makes her point or rolls 7. In this
latter case, we can use the geometric distribution on which governs the trial number of the first success in a sequence
of Bernoulli trials. The distribution of is a mixture of distributions.
32. The probability density function of is
Proof:
33. The first few values of the probability density function of are given in the following table:
1 2 3 4 5
0.33333 0.18827 0.13477 0.09657 0.06926
34. Find the probability that a game of craps will last at least 8 rolls.
Answer:
35. The mean and variance of the number of rolls are
a.
W
W
W
P(W = −1) =
10
11
P(W = 9) =
1
11
E(W ) = − ≈ −0.0909
1
11
sd(W ) ≈ 2.8748
W
W
N
N = 1
N+
N
N
P(N = n) = {
,
12
36
+ + ,
1
24
( )
3
4
n−2
5
81
( )
13
18
n−2
55
648
( )
25
36
n−2
n = 1
n ∈ {2, 3, …}
N
n
P(N = n)
E(N) = ≈ 3.3758
557
165
var( ) = ≈ 9.02376
245 672