2.6: Advanced Counting Using PIE

2.6: Advanced Counting Using PIE

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You have 11 identical mini key-lime pies to give to 4 children. However, you don't want any kid to get more than 3 pies. How many ways can you distribute the pies?

  1. How many ways are there to distribute the pies without any restriction?
  2. Let's get rid of the ways that one or more kid gets too many pies. How many ways are there to distribute the pies if Al gets too many pies? What if Bruce gets too many? Or Cat? Or Dent?
  3. What if two kids get too many pies? How many ways can this happen? Does it matter which two kids you pick to overfeed?
  4. Is it possible that three kids get too many pies? If so, how many ways can this happen?
  5. How should you combine all the numbers you found above to answer the original question?

Suppose now you have 13 pies and 7 children. No child can have more than 2 pies. How many ways can you distribute the pies?

Stars and bars allows us to count the number of ways to distribute 10 cookies to 3 kids and natural number solutions to (x+y+z = 11 ext{,}) for example. A relatively easy modification allows us to put a lower bound restriction on these problems: perhaps each kid must get at least two cookies or (x,y,z ge 2 ext{.}) This was done by first assigning each kid (or variable) 2 cookies (or units) and then distributing the rest using stars and bars.

What if we wanted an upper bound restriction? For example, we might insist that no kid gets more than 4 cookies or that (x, y, z le 4 ext{.}) It turns out this is considerably harder, but still possible. The idea is to count all the distributions and then remove those that violate the condition. In other words, we must count the number of ways to distribute 11 cookies to 3 kids in which one or more of the kids gets more than 4 cookies. For any particular kid, this is not a problem; we do this using stars and bars. But how to combine the number of ways for kid A, or B or C? We must use the PIE.

The Principle of Inclusion/Exclusion (PIE) gives a method for finding the cardinality of the union of not necessarily disjoint sets. We saw in Subsection how this works with three sets. To find how many things are in one or more of the sets (A ext{,}) (B ext{,}) and (C ext{,}) we should just add up the number of things in each of these sets. However, if there is any overlap among the sets, those elements are counted multiple times. So we subtract the things in each intersection of a pair of sets. But doing this removes elements which are in all three sets once too often, so we need to add it back in. In terms of cardinality of sets, we have

egin{equation*} |A cup B cup C| = |A| + |B| + |C| - |A cap B| - |A cap C| - |B cap C| + |Acap B cap C|. end{equation*}

Example (PageIndex{1}):

Three kids, Alberto, Bernadette, and Carlos, decide to share 11 cookies. They wonder how many ways they could split the cookies up provided that none of them receive more than 4 cookies (someone receiving no cookies is for some reason acceptable to these kids).


Without the “no more than 4” restriction, the answer would be ({13 choose 2} ext{,}) using 11 stars and 2 bars (separating the three kids). Now count the number of ways that one or more of the kids violates the condition, i.e., gets at least 4 cookies.

Let (A) be the set of outcomes in which Alberto gets more than 4 cookies. Let (B) be the set of outcomes in which Bernadette gets more than 4 cookies. Let (C) be the set of outcomes in which Carlos gets more than 4 cookies. We then are looking (for the sake of subtraction) for the size of the set (A cup B cup C ext{.}) Using PIE, we must find the sizes of (|A| ext{,}) (|B| ext{,}) (|C| ext{,}) (|Acap B|) and so on. Here is what we find.

(|A| = {8 choose 2} ext{.}) First give Alberto 5 cookies, then distribute the remaining 6 to the three kids without restrictions, using 6 stars and 2 bars.
(|B| = {8 choose 2} ext{.}) Just like above, only now Bernadette gets 5 cookies at the start.
(|C| = {8 choose 2} ext{.}) Carlos gets 5 cookies first.
(|A cap B| = {3 choose 2} ext{.}) Give Alberto and Bernadette 5 cookies each, leaving 1 (star) to distribute to the three kids (2 bars).
(|A cap C| = {3 choose 2} ext{.}) Alberto and Carlos get 5 cookies first.
(|B cap C| = {3 choose 2} ext{.}) Bernadette and Carlos get 5 cookies first.
(|A cap B cap C| = 0 ext{.}) It is not possible for all three kids to get 4 or more cookies.

Combining all of these we see

egin{equation*} |A cup B cup C| = {8 choose 2} + {8 choose 2} + {8 choose 2} - {3 choose 2} - {3 choose 2} - {3 choose 2} + 0 = 75. end{equation*}

Thus the answer to the original question is ({13 choose 2} - 75 = 78 - 75 = 3 ext{.}) This makes sense now that we see it. The only way to ensure that no kid gets more than 4 cookies is to give two kids 4 cookies and one kid 3; there are three choices for which kid that should be. We could have found the answer much quicker through this observation, but the point of the example is to illustrate that PIE works!

Pie Charts in Python

Plotly is a free and open-source graphing library for Python. We recommend you read our Getting Started guide for the latest installation or upgrade instructions, then move on to our Plotly Fundamentals tutorials or dive straight in to some Basic Charts tutorials.

A pie chart is a circular statistical chart, which is divided into sectors to illustrate numerical proportion.

If you're looking instead for a multilevel hierarchical pie-like chart, go to the Sunburst tutorial.

Pie chart with plotly express¶

In px.pie , data visualized by the sectors of the pie is set in values . The sector labels are set in names .

Pie chart with repeated labels¶

Lines of the dataframe with the same value for names are grouped together in the same sector.

Pie chart in Dash¶

Dash is the best way to build analytical apps in Python using Plotly figures. To run the app below, run pip install dash , click "Download" to get the code and run python .

Get started with the official Dash docs and learn how to effortlessly style & deploy apps like this with Dash Enterprise.

Setting the color of pie sectors with px.pie¶

Using an explicit mapping for discrete colors¶

For more information about discrete colors, see the dedicated page.

Customizing a pie chart created with px.pie¶

In the example below, we first create a pie chart with px,pie , using some of its options such as hover_data (which columns should appear in the hover) or labels (renaming column names). For further tuning, we call fig.update_traces to set other parameters of the chart (you can also use fig.update_layout for changing the layout).

Basic Pie Chart with go.Pie¶

If Plotly Express does not provide a good starting point, it is also possible to use the more generic go.Pie class from plotly.graph_objects .

In go.Pie , data visualized by the sectors of the pie is set in values . The sector labels are set in labels . The sector colors are set in marker.colors .

If you're looking instead for a multilevel hierarchical pie-like chart, go to the Sunburst tutorial.

Styled Pie Chart¶

Colors can be given as RGB triplets or hexadecimal strings, or with CSS color names as below.

Controlling text fontsize with uniformtext¶

If you want all the text labels to have the same size, you can use the uniformtext layout parameter. The minsize attribute sets the font size, and the mode attribute sets what happens for labels which cannot fit with the desired fontsize: either hide them or show them with overflow. In the example below we also force the text to be inside with textposition , otherwise text labels which do not fit are displayed outside of pie sectors.

Controlling text orientation inside pie sectors¶

The insidetextorientation attribute controls the orientation of text inside sectors. With "auto" the texts may automatically be rotated to fit with the maximum size inside the slice. Using "horizontal" (resp. "radial", "tangential") forces text to be horizontal (resp. radial or tangential)


Type 1 diabetes mellitus (T1DM) is one of the most challenging medical disorders, and one of the key therapeutic goals to prevent or delay long-term diabetes complications in T1DM is the achievement and maintenance of near-normal glycemic control 1 . Only insulin treatment is not enough to rely on, dietary adjustments also play an important role in the regulation of blood glucose. Carbohydrates are a major determinant of postprandial blood glucose. Carbohydrate counting is a meal planning approach used with clients who have diabetes that focuses on carbohydrate as the primary nutrient affecting postprandial glycemic response. Advanced carbohydrate counting allows adjustment of the prandial insulin dose for actual carbohydrate intake in T1DM patients on intensive insulin therapy. Therefore, by calculating the carbohydrate amounts in each meal, insulin doses required to preserve postprandial blood glucose within normal limits can be predicted 2,3 . The current guidelines recommend that the algorithms for prandial insulin calculation take into account the carbohydrate amount of the meal 4 .

However, the efficacy of carbohydrate counting is not fully understood. At present, study to inquiry the effect of carbohydrate counting on T1DM patients is relatively little. Evidence from randomized controlled trials (RCTs) reported inconsistent results 5,6,7 . Previous systematic review 8 or meta-analysis 9 did not cite a clear conclusion. Furthermore, three recent trials 10,11,12 with adequate power have been published and involve new evidence. Therefore, we performed this meta-analysis to evaluate the efficiency of carbohydrate counting on glycemic control in people with T1DM.

18 Electron Rule

Coordination complexes most often have 18 electrons on the metal atom because eighteen is the number of electrons in the nearest noble gas configuration in transition metals, which includes s + p + d electrons. The second and third row of transition metals have f electrons, too, but we usually simplify and don't worry about those electrons for electron counting purposes we treat them like core electrons, not valence electrons. Frequent exceptions include 16 electron complexes, which are especially common in late metals such as copper, nickel and palladium. There are molecular orbital considerations that contribute to these exceptions.

Occasionally, electron counts can be as low as eight electrons. These cases often occur in early transition metals, such as titanium or tantalum. These metals have a long way to go to get to eighteen electrons, and sometimes they cannot fit that many ligands in their "coordination sphere". Hence, steric factors can prevent metals from reaching eighteen electrons.

Figure CC3.5. (Silox)3Ta, or (tBu3SiO)3Ta, an electronically unsaturated metal complex.

Complexes that do not have eighteen electrons are sometimes called "electronically unsaturated". Like six electron complexes in main group chemistry, they often react with donors in order to increase the electron count at the metal atom.

Deconstruct (silox)3Ta into ligands and metal atom (or ion) and confirm the electron count on tantalum.

Because steric crowding is one of the reasons that metals in coordination complexes may remain "electronically unsaturated" (meaning not enough electrons) or "coordinatively unsaturated" (because there aren't enough donors), it is important to pay attention to how much space a ligand will occupy. The parameter used to describe this factor is called a "cone angle". This is just the angle at the nose of the imaginary cone where the ligand attaches to the metal. The breadth of the cone, and thus the cone angle, is detarmined by how large the ligand is. A large cone angle indicates a large ligand a small cone angle is associated with a smaller ligand.

For example, trimethylphosphine, P(CH3)3, has a cone angle of 1180. Triethylphosphine, P(CH2CH3)3, is slightly larger, with a cone angle of 1320.

Create a Power BI Pie chart

Creating a Power BI Pie chart can be done in just a few clicks of a button. But then there are some extra steps we can take to get more value from it.

The purpose of a Pie chart is to illustrate the contribution of different values to a total.

For example, to see the total sales split by product category. You can then see the percentage contribution of each product category to the total revenue.

The Pie chart is not the only chart type that can produce this visual. We could opt for a Donut chart, or maybe a Column chart instead.

It is important to be able to create different chart types, as you may be asked by someone to display data in a specific way.

Check out this quick video on how to create a Power BI Pie chart:

1. Navigate to Report View

You need to be in report view to create your visualizations.

If you are not there already, you need to navigate to this view by clicking the Report button along the left side of the window.

After navigating to this view, it is a good habit to click somewhere within the white area. This ensures that no existing visualizations are selected when we go to create our new one.

2. Add the Pie chart to the report

Click the Pie chart button in the Visualizations area.

A Pie chart is added to the report page. More options have also appeared in the Visualizations pane. We will be interacting with these next to specify what data should be displayed and how.

Click the arrow to the left of the Products and Sales tables to expand the list of fields.

Using fields in Power BI visualizations is similar to how you work with PivotTables in Excel.

Drag the Product Category field from the Products table into the Legend area, and the Total Revenue measure from the Sales table into the Values area.

The following Pie chart is created.

You can resize the chart using the handles around the edge of the visual. You can also move the chart on the page.

3. Basic formatting

There are plenty of formatting options that can be applied to a Pie chart to improve it.

Click the Format button to access all the formatting options for the selected visualization.

Examples of some useful changes could be to increase the text size of the data labels (Detail Labels section) and chart title (Title section).

There are tons of options and it is really a case of looking around and seeing what is available, and what you like to use.

You can see some more examples of formatting changes in the video at the end of this article.

4. Mouse over for more statistics

A cool effect with Power BI Pie charts is that when you mouse over the different segments, a tooltip with statistics is shown.

You can add more information to these tooltips if you want by dragging fields into the Tooltips area.

If you have just been making some formatting changes, you will need to click the Fields button first to switch back to field options.

Click and drag the Average Sales Total measure into the Tooltips area.

This information is then displayed in addition to Total revenue in the tooltip.

You could go on to add further information to tooltips if required.

5. Adding to the Details area

By adding fields into the Details area of a Pie chart we can create an even further breakdown of our data

Click and drag the Years field into the Details are of the visualization.

This can be a cool feature and show some great insights into our data.

In this example though, it also highlights some limitations with Pie charts. This visual is showing too much data and it is difficult to see what we want.

If we were to keep this visual, we would look to turn the Legend back on as it was lost when the Details area was used.

We will remove the Years field from the details area.

The chart title is becoming confusing. Let&rsquos change this in the Format options for the chart.

In an example such as this. A better idea may be to create a second visualization and have them interact with each other.

6. Interactive visualizations

Let&rsquos add another visualization to the report - the Matrix. This visualization is similar to a PivotTable.

1. Click in the white space of the report page to deselect the Pie chart.

2. Click the Matrix button in the Visualizations pane.

3. Expand the salesReps table and drag the Sales Rep field to the Rows area of the Matrix.

4. Expand the Sales table and drag the Total Revenue field to the Values area, and the Year field into the Columns area of the Matrix.

The Matrix will look like below. This is a nice visual of each sales rep performance over the years.

The really cool thing now, is that the Pie chart visualization will interact with the Matrix and vice versa.

Click a product category segment in the Pie chart and see how it filters the Matrix.

In the image below, the Meat/Poultry data point has been selected in the Pie chart. The Matrix now only shows the values for that product category.

And if you click a value in the Matrix, it will visualize that selection in the Pie chart.

In the image below, the total value for Nancy Davolio has been selected.

With the data in use here, the visual on the Pie chart is not very useful. It is so small, we cannot read much into it. But this could be a really effective visualization.

However, the filter from the Pie chart to the Matrix is great.

To clean a filter, click somewhere in the empty space of the visualization that you clicked on.

These interactions can be edited so that the report works as we desire.

We would like to keep the filter from the Pie chart to the matrix, but remove the one to the Pie chart.

Select the matrix visualization. Then click on the Format tab and the Edit Interactions button.

This will display the interaction buttons for the other visualizations on the report page.

Now if we wanted to disable all interactions to the Pie chart from the matrix we would click the no impact icon.

Click on the Highlight button to disable the visual interaction to the Pie chart.

Although we do not want the Highlight interaction, having the matrix filter the Pie chart would be great.

With the matrix selected, click the Filter button for the Pie chart to toggle the filter functionality on.

Below is an image of the Pie chart being filtering to show Nancy Davolio&rsquos sales only.

The interactions between the Pie chart to the matrix remain the same.

CVData Features

  • Rounds - Up to 300,000,000,000 rounds of Blackjack can be simulated.
  • Players - 1-7 seats, each seat can use a different playing & betting strategy. Also, you can vary the number of players at the table during a sim.
  • Data - I figure, if you're going to play billions of hands, you ought to record what happened. Up to 20,000 stats are created per seat.
  • Reports - Printing all of this data requires 250 pages. Clearly this is silly. So, a reporting subsystem has been included. A customized report can be defined including only the data that you wish for the players that you wish. To deal with the large variety of stats descriptions can be included on the reports. Nearly all printers are supported and formatted tables with proportional fonts are used to improve readability.
  • WWW Publishing - But, what about the Internet. The same reporting facility can be used to create a customized report for Web publishing. The data will be written to disk using HTML tables and formatting for readability. This file can be displayed and edited with a browser or posted to a web page. See a sample one seat WWW report.
  • MS Word - Alternatively, you can create a Microsoft Word document which you can then edit.
  • Charts - 41 standard graphs.
  • Do it yourself Graphs - I find that graphing sim data greatly improves understanding of results. With all this data, I could not decide what to include in graphs in addition to the 41 standard graphs. So, a Do-it-yourself graphing subsystem has been added. You can take any sets of the 60,000 data items and graph them. You can also take data from different sim runs and graph them together. In fact, you can take data from different sims run on different PC's in different countries and graph them on one chart. A separate help file has been included to describe the hundreds of graph options.
  • CVSpread - A spreadsheet subsystem has been included with CVData that can support Excel commands, functions and files. 143 math functions are included.
  • Number formatting - To ease readability, thousands separators are now included in the 60,000 stats. To support the huge number of rounds, each number is tested for size and converted to scientific format if necessary. To improve readability for non-U.S. users numbers, currencies and dates are customized by country.
  • High-Resolution Monitors - As most PC's are now delivered with 800x600, 1024x768 or higher resolution. These resolutions are now supported to display more card counting data at once without scrolling and more digits of significance. The conversion to scientific format will be at higher thresholds with larger screen resolution.
  • Playing Strategies - CVData can use the 400 CVBJ card counting tables and strategy creation screens.
  • Betting strategies - A new, much simpler, method of specifying a betting strategy has been included. Although it is much easier to use, it still provides for sophisticated camouflage plays and betting errors. I believe that a sim without sophisticated cover play simulation is useless. For more advanced strategies (wonging in and out, advanced camouflage, combination. counting./progression for camouflage, etc.) the old advanced method of defining a betting strategy is also included. For details see Blackjack Betting Strategies.
  • Wonging/Backcounting - Up to six different wong-in and separate wong-out points can be specified per seat by depth. You can also specify when you will stop observing a table and move to the next by depth. Wonging-cover alsosupported.
  • Optimal Bet calculation - This feature supports optimal bet calculations for 1-7 hands and changing number of hands by the count. Bets can be fine-tuned after the sim and some statistics will be recalculated.
  • True-counting Unbalanced Strategies - Strategies like TKO (true-counted KO) are supported for simulation and index generation.
  • Usability Features - Such as:
    • Batch facility
    • Multi-Sim. This feature allow simple execution of a very large number of sims varying up to four parameters.
    • Halves Counts can be accurately simulated.
    • Double-barrier RoR/Bankroll calculator
    • Trip Ruin Calculator
    • Modern Windows interfaces
    • Different betting strategies by deck depth or bankroll
    • Different playing strategies by deck depth, bankroll or number of cards in your hand
    • Multi-Parameter strategies
    • Counting by Inference simulation
    • Counting by Inference strategy creation
    • Insurance side count
    • Peeking at other players' cards
    • Lots of new, unique options for customizing deck estimation and True Count calculations.
    • Ability to assign different strategies to each player and collect statistics for all players. Each of the seven players can be assigned up to six playing and six betting strategies switching strategies at different deck depths.
    • Betting errors
    • Strategy errors
    • Multiple hands with different numbers of units.
    • Double Down Rescue . One strategy is used for normal Surrender and different strategy for a Surrender after a Double.
    • Discard after Split supported allowing you to throw away a split hand. A separate strategy is supported to indicate when to discard the Split hand.
    • Discard after Double supported. A separate strategy is supported to indicate when to discard the DD card and draw another.

    Pie Chart

    Pie charts are used to display the contribution of each value (slice) to a total (pie). Pie charts always use one data series.

    To create a pie chart of the 2017 data series, execute the following steps.

    2. On the Insert tab, in the Charts group, click the Pie symbol.

    4. Click on the pie to select the whole pie. Click on a slice to drag it away from the center.

    Note: only if you have numeric labels, empty cell A1 before you create the pie chart. By doing this, Excel does not recognize the numbers in column A as a data series and automatically creates the correct chart. After creating the chart, you can enter the text Year into cell A1 if you like.

    Let's create one more cool pie chart.

    5. Select the range A1:D1, hold down CTRL and select the range A3:D3.

    6. Create the pie chart (repeat steps 2-3).

    7. Click the legend at the bottom and press Delete.

    9. Click the + button on the right side of the chart and click the check box next to Data Labels.

    10. Click the paintbrush icon on the right side of the chart and change the color scheme of the pie chart.

    11. Right click the pie chart and click Format Data Labels.

    12. Check Category Name, uncheck Value, check Percentage and click Center.

    Note: right click the data labels and click Font to change the font size and font color of the data labels.

    Proof of sustainability as the basic principle

    Companies must specify the feedstock from which the biofuel has been produced when submitting their claim. They must base this on the feedstock as stated in the Proof of Sustainability or the Guarantee of Origin (the latter in the form of a Vertogas certificate).

    The Energy for Transport Registry (REV: Register Energie voor Vervoer) contains a list of feedstocks as used in sustainability systems (ISCC in particular). If the template for the Proof of Sustainability does not include a specific feedstock for a claim, companies can contact their voluntary scheme or NEa.

    Robot Framework User Guide is a reference manual explaining all Robot Framework features in detail. You can either View the selected version online or Download it as a zip package. This package contains also standard library documentation under libraries directory.

    These test libraries are distributed with Robot Framework. Click View to view the selected version online, and use Ctrl-S or equivalent to save the opened page locally if needed

    New standard libraries are added time to time. Dropdown menus list versions in which libraries are available.

    BuiltIn View Contains generic often needed keywords. Imported automatically and thus always available.
    Collections View Contains keywords for handling lists and dictionaries.
    DateTime View Supports creating and verifying date and time values as well as calculations between them.
    Dialogs View Supports pausing the test execution and getting input from users.
    OperatingSystem View Enables performing various operating system related tasks.
    Process View Supports executing processes in the system.
    Remote N/A Part of the remote library interface. Does not have any keywords of its own.
    Screenshot View Provides keywords to capture and store screenshots of the desktop.
    String View Library for manipulating strings and verifying their contents.
    Telnet View Supports connecting to Telnet servers and executing commands on the opened connections.
    XML View Library for verifying and modifying XML documents.

    Omega 2 / Omega II Blackjack Card Counting System

    The Omega II count system is another one of the most powerful card counting strategies in blackjack. It is relatively new and was made famous in a book called "Blackjack for Blood" by Bryce Carlson. This book was published in 2001 so it was not too long ago. The basics of the of the Omega 2 system start with a multi-level count style, so it is a little more advanced than other systems. In fact, this is a level 2 advanced variety, so it has a moderate difficulty level and it isn't overly complicated.

    Omega II is extremely efficient and accurate. In fact, the efficiency level for betting is close to 99%, which is a nearly perfect system out there and you are getting essentially the largest advantage you can possibly get in any card counting strategy. Surprisingly, Omega II is very similar to the Zen count and virtually identical to the Canfield Master in terms of the indices used and the counting values used for each card. The only difference is now the zeros are neutral and the 9s are assigned values of -1 for the count.

    The distribution of assigned point values for each type of card are shown below. The column to the left shows the cards from two through Ace and the column to the right shows what count is assigned to each of these cards. This is known as the indices for the Omega II card counting system.

    Card Value Points
    2 +1
    3 +1
    4 +2
    5 +2
    6 +2
    7 +1
    8 0
    9 -1
    10 -2
    J -2
    Q -2
    K -2
    A 0

    Note that the level 2 refers to the wide variety of count values assigned to the cards. A level 1 basic system only has the count values (-1,0,+2) but the Omega II has the values ranging from (-2,-1,0,+1,+2). This is what makes this system more complicated than others because you have to remember a wider range of count values for each of the cards.

    The Omega II blackjack system is also a balanced system, meaning that you start your count off at 0 when the deck is first dealt and the ending count should also be 0 after the deck or shoe of cards have all be dealt out. If your running count does not start and end at zero when the dealer goes through the entire deck, then you know that you have lost track of the correct count. This is sort of beneficial if you are just learning how to count cards though.

    Remember, the aces are counted as 0 points so you can actually optimize this system even further. Generally when the aces are counted as null or zero, a player should be keeping track of the aces in a separate count off to the side. This quickly becomes a complex system by trying to keep track of two separate counts. If you do decide to count the aces and you think you have enough experience, then you should be comparing how many have been dealt to how many are still in the deck in order to make even better decisions.

    When using the Omega 2 system, the goal is to wait for a high positive count just like most other card counting methods. When the count is high, this means many low valued cards have been dealt and there are many high cards still left to be dealt. Likewise, when the count is low, there are many low cards still in the deck and the high 10 point cards were already used up. The strategy is to bet high when the count is high and make low wagers when the count is low (especially when negative).

    Watch the video: Advanced Counting Using The Principle Of Inclusion And Exclusion (August 2022).