Dice, Cards, and Probability part 2 Linear Results

In Part 1, I looked at probability, notation, number of occurrences 

Linear Die/Dice (single) Results

Looking at single die/dice rolls, which are linear in results. Linear means that there is a set incremental probability, which if graphed would form a straight line. Rolling a single die gives linear results. As opposed to rolling multiple dice which would create a curvilinear result, normally more triangular or in a bell curve shape.

Important probability increments

Fifty-fifty

The first natural probability divide is half. Fifty-fifty chances, much like the flip of the coin, is a common and popular. All of the dice except d3 can show fifty-fifty percent rolls, since the rest are multiples of two. These are d6, d8, d10, d12, and d20

Rule of thirds

Sometimes things aren't just "yes or no" or "black and white". Sometimes you need "maybe" and "gray/grey". Those dice divisible by three can show such results excactly. These are d3, d6, and d12.

Quarters

Another popular divisible chance is in quarters. Number of dice divisible by four can show such results exactly. These are d8, d12, and d20.

Dice Probability d3, d6, d8, d10, d12, and d20

# number, #% percentage of rolling that number, >% percentage of rolling greater than that number, >=% percentage of rolling greater or equal to that number.

d3
#	#%	>%	>=%
1	33%	67%	100%
2	33%	33%	67%
3	33%	0%	33%

d6
#	#%	>%	>=%
1	17%	83%	100%
2	17%	67%	83%
3	17%	50%	67%
4	17%	33%	50%
5	17%	17%	33%
6	17%	0%	17%

d8
#	#%	>%	>=%
1	13%	88%	100%
2	13%	75%	88%
3	13%	63%	75%
4	13%	50%	63%
5	13%	38%	50%
6	13%	25%	38%
7	13%	13%	25%
8	13%	0%	13%

d10
#	#%	>%	>=%
1	10%	90%	100%
2	10%	80%	90%
3	10%	70%	80%
4	10%	60%	70%
5	10%	50%	60%
6	10%	40%	50%
7	10%	30%	40%
8	10%	20%	30%
9	10%	10%	20%
10	10%	0%	10%

d12
#	#%	>%	>=%
1	8%	92%	100%
2	8%	83%	92%
3	8%	75%	83%
4	8%	67%	75%
5	8%	58%	67%
6	8%	50%	58%
7	8%	42%	50%
8	8%	33%	42%
9	8%	25%	33%
10	8%	16%	25%
11	8%	8%	16%
12	8%	0%	10%

d20
#	#%	>%	>=%
1	5%	95%	100%
2	5%	90%	95%
3	5%	85%	90%
4	5%	80%	85%
5	5%	75%	80%
6	5%	70%	75%
7	5%	65%	70%
8	5%	60%	65%
9	5%	55%	60%
10	5%	50%	55%
11	5%	45%	50%
12	5%	40%	45%
13	5%	35%	40%
14	5%	30%	35%
15	5%	25%	30%
16	5%	20%	25%
17	5%	15%	20%
18	5%	10%	15%
19	5%	5%	10%
20	5%	0%	5%

Deck of Cards

Also, I look at what a person can do with a standard 52 card deck when shuffled. A disadvantage of the card deck is that to retain the probability, one must reshuffle, or the probability changes each additional drawing of cards.

I also notice that one can use a 12 sided die/dice and nearly get the same probability, since the difference between 12 and 13 chances is very minimal.

Standard Playing Card 52 card deck (French suit)
Action		#%	>%	>=%
Picking one specific card	1/52	2%
Picking one number or face	1/13	8%
1 (ace)		8%	92%	100%
2		8%	85%	92%
3		8%	77%	85%
4		8%	69%	77%
5		8%	62%	69%
6		8%	54%	62%
7		8%	46%	54%
8		8%	38%	46%
9		8%	31%	38%
10		8%	23%	31%
jack		8%	15%	23%
queen		8%	8%	15%
king		8%	0%	8%
Picking a face [jack, king, queen]	12/52	23%
Picking one suit [spade, club, heart, diamond]	1/4	25%
Picking a red (or black)	1/2	50%
Picking a non-face	40/52	77%

Comparing Dice

Here is a linear version, rounded to multiples of 5, comparing all six dice.

Dice Comparison Probability
Die/Dice	5%	10%	15%	20%	25%	30%	35%	40%	45%	50%	55%	60%	65%	70%	75%	80%	85%	90%	95%	100%
d3	33%							67%						100%
d6	17%			33%				50%			67%			83%			100%
d8	13%			25%		38%			50%		63%			75%		88%			100%
d10	10%		20%		30%		40%		50%		60%		70%		80%		90%		100%
d12	8%		16%	25%		33%		42%	50%		58%		67%	75%		83%		92%	100%
d20	5%	10%	15%	20%	25%	30%	35%	40%	45%	50%	55%	60%	65%	70%	75%	80%	85%	90%	95%	100%

Thoughts on d100

I did consider d100 briefly, but then realized something. Looking over most of the bonuses of the d100 systems, they seem to change by +/-5. If most d100 systems ignore bonuses not multiple of 5, that tells me that going below 5 will not really affect the final result sufficiently. A +1 for a d100 isn't really much of a bonus to affect the outcome. In a sense, they seem like d100/5 systems - basically an exploded d20 systems.

Conclusion

I really like d12

Although d20 is the die/dice of the more popular games, I am seeing d12 as a very good one which can also do the rule of thirds type results, as well as quarters and halves. Using d12 also somewhat mimics a deck of cards.

I had never considered using a d12 as a main part of a system. However, looking over the probability, I may prefer d12 over d20. Doing so may also help to crush down the run-away bonus/penalties to more like +/-3 ranges much more like Fate/Fudge.

In the next dice post Curvilinear or Bell Curve Results. 

Quick free world maps using GIMP filters and Donjon fractal maps - Part 5: Expanding the Zoom limits

Quick free world maps using GIMP filters and Donjon fractal maps series:

Part 1: Getting a fractal map,

Part 2: Getting different types of maps,

Part 3: More accurate 3D maps with mountains,

Part 4: Zooming and resolution,

Part 5 Expanding the Zoom Limits.

Part 6: Quick outline maps

Available from Google Play Books.
https://play.google.com/store/books/details/Ken_N_Wickham_Quick_free_world_maps_using_GIMP_fil?id=tvZmDAAAQBAJ

Also available on Kindle
https://www.amazon.com/Quick-world-filters-Donjon-fractal-ebook/dp/B01H131XPY/

http://www.rpgnow.com/product/187006/Quick-free-world-maps-using-GIMP-filters-and-Donjon-fractal-maps

Expanding the Zoom Limits

This post continues zooming in closer in the same section as Part 4. It will zoom in even closer. For more information about the region, see part 4.

Chunk of original map

This is the original world map chunk for the zoomed in area. It is 40 pixels height by 40 pixels width. It is approximately two hexes big. In the hex map, you can see that the hexes where it is located show the symbols of mountains. I added the snow mountain symbol.

Hex map. The zoomed area is the snow mountains in the lower right. They are both at the same scale.

Problem with chunk; pixelation

Zoomed in, the information from the original square looks like this. You can actually count the square pixels. Sure you can sort of see valleys and hills in this picture. But I like doing things easy and fast if possible. Yet also maintaining somewhat accurate information. Digging more information from this small chunk quick, easy, and accurate would be nice.

Greyscale 3D map

This is the best view of that area using the newest 3D information that I showed you in post 3. The pixels are very apparent. 

My technique

This sort of was the area that I was focusing on. Later I decided to move it a little more to the right and down more to capture the entire mountain range. You can see the topographical contour lines

A more accurate map with detailed contour lines

Here is that information using my own technique. After finding this technique out accidentally, I recognized the great value in having a quick method to generate a fairly accurate topographical map with contour lines. I also calculated that each line equaled 100 foot elevation change increments. You can see that post here where I count from sea level up to some snow peaks.

I really like this map.

In a prior post and different area near water, I also explored using the contour lines to add smaller rivers. I also tried drawing the hills using the information from the original map. You can see a comparison to old methods and the results of my hand-drawn rivers and lakes in the post here.

Tool to help determine raised and recessed areas

This map helps interpret the contour lines. The light is coming from the northwest corner. Contour line areas with light on the northwest are raised. Areas with dark on the northwest are recessed. I make a few notes concerning different features.

Zooming into the three main peaks

For even closer detail I focus on just the highest peaks, the ones I labelled Twin Peak and Mountain Peak.  I did one for contour and the 3D version. Click on each to see the full image. The original of the next two images is 17 x 9 pixels big. Even then, a lot of information can be extracted from that fragment of the map.

Original Chunk

Original image chunk =>

The little chunk of image above, which is smaller than many words in this sentence, is what was used to create the next two images. This area is less than one hex big. So we are actually looking inside one of the hexes, to gather more information. This information can be used in describing the game world with even more detailed description. It is more like an adventure map were you can place individual buildings, caves, ruins, camps, and sites.

All from the same original fractal map.

Topographical Contour Line map of peaks

For full image, click on the image.

3D version showing raising or recessed elevation

For full image, click on the image

I can see that the peak on the left has two bumps and the one on the right is more pointed, as well as slightly separated from the rest of the slope, sitting more on a shelf. I can imagen that the areas between are great glaciers carving out the valleys and melting into streams and rivers.

Next Month

Next month I will try posting on how to make quick outline maps of the world. Outline maps good for printing off and hand writing in region areas or giving to players. 

Quick free world maps using GIMP filters and Donjon fractal maps - Part 4: Zooming and Resolution

Quick free world maps using GIMP filters and Donjon fractal maps series:

Part 1: Getting a fractal map,

Part 2: Getting different types of maps,

Part 3: More accurate 3D maps with mountains,

Part 4: Zooming and resolution,

Part 5 Expanding the Zoom Limits.

Part 6: Quick outline maps

Available from Google Play Books and RPGNow.
https://play.google.com/store/books/details/Ken_N_Wickham_Quick_free_world_maps_using_GIMP_fil?id=tvZmDAAAQBAJ

Also available on Kindle
https://www.amazon.com/Quick-world-filters-Donjon-fractal-ebook/dp/B01H131XPY/

http://www.rpgnow.com/product/187006/Quick-free-world-maps-using-GIMP-filters-and-Donjon-fractal-maps

Zooming and Resolution

This full world map of Aioskoru shows the outline of the mountainous region that I wanted to test methods out on. I really wanted to find a method to quickly generate topographical maps showing contour lines, especially if I could do so at a more finely detailed resolution.

I show the series of attempts, failures, and breakthroughs that increased my knowledge of methods of map information extraction digital methods using fractal maps.

This next picture is the Plain Color Atlas zoomed image of area. The other images following will be of this same region. You can compare each of the methods results to each other. In this picture the whiter areas are snowcapped peaks.The browns are the regular mountains. The other areas are valleys.

This was one of my first attempts at trying to modify and bring out more information from the picture. The result added structure, but wasn't very good nor accurate.

Here is the default Icon of the region. Certainly the icons are symbolic on a grand scale and not so accurate on a closer or fine detail sense.

In this view, I used the icon pack to modify several of the mountains into the snowcapped peak icon. I also tried to stack the grid on top of the first modification of the map. For a time, this became my most accurate map. It still is my main hex map.

This was the result of my first attempt at a topographical map with contour lines. The fractal is very obvious and unusable, but I realized that this meant something. if only I could harness this information better.

I then found this method to separate the layers of the mountains. This method became what I call the Layered Method. I had actually tried to use the color selector to make this similar type of map, before finding this quicker one step method. That other method took 5 minutes to select each of the different color ranges in the entire map. Even then, I missed some. This one step method is better and very easy.

With the Bump tool, this became my first 3D map using the Layer map. I quickly realized that the mountains were depressed down rather than sticking up.

I used a select and reverse method to try pulling up the mountains. This for several months remained my best 3D map. This was going to be the method I was going to share when I wrote my first blog post, up until I found the quicker method more recently. It takes time, patience, and luck to get this type of result. Even then it is somewhat flawed.

Here is the Greyscale method that I recently found and shared. I believe it is easier and more accurate than that last complex method and doesn't have the weird indentation flaws. When combined with the Plain color map like I showed in the previous post, it gives a pretty decent and accurate map.

For a month I assumed that that method was the best that I could do for more accurate detailed information. It isn't bad. But of course I wanted more. While working with the map, I was trying to do something entirely different than typography. This emerged accidentally. It has something to do with that black map with mutliple color lines.

When I looked at it closely, I realized that it shows topographical lines showing changes in elevation at a more finely resolution. And I thought that I could zoom in at a finer detail. 

In the next post I will show a comparison of that zoomed in area.

Here is a composite picture mixing the Greyscale and the Topographical Contour Line method maps. I noticed that I could see smaller hills and valleys easier. Elevations changes that did not even show up on the 3D Grayscale Bump map indicated by a lighter or darker circles and patches of color. 

At this time, I wont show that the Topographical Contour Line method. That method I may actually put in a PDF book in the future and sell for a small cost. Or I may offer that service for a fee for select regions of someone's maps, since that method took a lot of frustration, attempts, and failures. And only an accident while painstakingly working with the maps revealed it. It took so long to develop.

The next post next week, I will show the close up zoomed comparison, even more finely detailed than this one zooming in on that southern mountain range slope.

Dice, Cards, and Probability

Part 1 - Dice, Cards, and Probability - Foundation

This blog post is the first for a possible series of posts concerning dice and card probability. While I do not present any new information, I am posting this so that the basics will be covered.

I plan on looking at d20, d6, d100, and card decks in the future.

What is probability?

Probability is the chance or likelihood that an event or situation will happen or occur.

How do you simulate random occurrence in gaming?

In RPGs, probability and random occurrence is simulated through multiple means, methods, tool, or medium that a random probability is generated, called Random Number Generator (RNG) in computer science. Dice are the most commonly used tool to generate random numbers. Two other popular physical RNGs are a deck of cards and a computer program, app, or website. Coins, rock-paper-scissors, or guessing a number between _ and _ are other less popular methods.

Expressing the probability of single occurrence using notation.

In math, basic probability can be expressed in terms of "how many times occurrences" out of "how many times attempted".

let # = number
let O= # of occurrences
let T= total number of tries

successful O out of total T
"out of" keyword O out of T
Colon O:T
Division notation O/T or O÷T. It's best to simplify fractions.

Since probability can be expressed as a division, you can find a decimal between 0 and 1 by dividing the total number into the number of occurrences.

And that number can be converted into a percentage (%).

An example of expressing the probability of getting 'heads' on a coin flip.

For example: A coin flip is either heads or tails.
The chance of getting heads is one occurrence out of two: either heads or tails.
1 out of 2
1:2
1/2
To convert the chance of
1/2 or 1 ÷2 = 0.5
0.5=50% chance

Combinations Formula

! = factorial symbol
let n = number of items
let r = number of items being chosen at a time
nCr = n!/r! * (n-r)

Quick free world maps using GIMP filters and Donjon fractal maps - Part 3: More Accurate 3D mountain maps

Quick free world maps using GIMP filters and Donjon fractal maps series:

Part 1: Getting a fractal map,

Part 2: Getting different types of maps,

Part 3: More accurate 3D maps with mountains,

Part 4: Zooming and resolution,

Part 5 Expanding the Zoom Limits.

Part 6: Quick outline maps



Available from Google Play Books.
https://play.google.com/store/books/details/Ken_N_Wickham_Quick_free_world_maps_using_GIMP_fil?id=tvZmDAAAQBAJ

Also available on Kindle and RPGNow
https://www.amazon.com/Quick-world-filters-Donjon-fractal-ebook/dp/B01H131XPY/

http://www.rpgnow.com/product/187006/Quick-free-world-maps-using-GIMP-filters-and-Donjon-fractal-maps

Part 3: A More Accurate 3D world map with mountains

In the first tutorial in part 1 (here), I showed you how to find a map, record the seed #, change the water and ice levels, make layer, and make 3D top perspective maps using GIMP.

In part two (here), I showed you how to use that seed and find different types and views of maps.

I said in the first part that I would post later a better way to make 3D maps that overcomes the brown mountain inverse problem. I was going to show you that method now, but since that post I found a more accurate and better way to do so. So instead, I will show you the new method of making a more accurate 3D world map that has correct mountains.

1. Go to Donjon fractal world generator. http://donjon.bin.sh/world/ 
2. Enter the seed # from part 1 and part 2.
3. Make sure the ice and water percentage are the same. The water and ice won't matter in the map you will be making, but you are just trying to make sure that you have the same map.
4. Change the Iteration to 25000 instead of 5000 default as you did in part 2.
5. You should see the same world now in the preview view if you click on an empty spot outside the Iteration box so it can update the world.
6. In the Map Palette box, select Greyscale instead of Atlas.
7. You can pick the height of the map that you want, the default being 400 pixels. The largest being 2000 pixels. Pick the same Image Height size as the Plain atlas map, since you will be combining both maps. Make sure they have the same Rotation as well. 
8. Save the greyscale map. I like calling it grayscale or greyscale, depending on one's preference for the word gray/grey. As long as you know what it means.

In order to use this map, you will need to save the GIF as a JPEG as well to use in the next step. Open the GIF of greyscale in a basic image program such as Paint, and save it as a .jpg or jpeg type image file.

With this map, we can turn the greyscale map into a 3D map much like we did in part 1

1. Open the JPEG version of the map using GIMP. 

2. Go to Filters > Map > Bump Map

3. Make sure first that the Bump Map is the same map as your current map. When working with several maps, it may show another map as the bump map source. In those cases, the bump texture won't match the the actual picture.

 Elevation: I like at 0.5 or 1.5.

 Depth: I like at 1 or 2

Azimuth changes which direction the sun is coming from and the opposite side will cast shadows. You can see

Don't offset the bump map.

4. Save this map. I normally call it the name_of_the_world_3D map.

If you notice the map now, there is definitely elevation texture. It is missing however color and water. For that, we can layer the Plain map over this texture map.

1. Open the Greyscale 3D map saved, using GIMP, if you don't have it open still.
2. Open the Plain Color  atlas map pallette map of the same size, which should be the same type as the first map you saved in part one. Make sure that you convert the Plain atlas color image to a JPEG as well using the same process.
3. Edit > Copy the Plain Color map. 
4. Switch to the Greyscale 3D map.
5. Edit > Past As > New Layer to add the Plain color map and water/ice to the Greyscale version in different layers.
6. Make sure you have the layer window. Windows > Dockable Dialongs > Layers  or CTRL + L.
7. You should see both pictures as two different layers. Rename the layers by double-clicking on the name of the layer, so that you know what that layer is. I like Plain and 3D Greyscale/Grayscale (whichever you preference for the word grey/gray.
8. 
9. Single-click on plain, which should be on top. Notice the Opacity is now at 100%.
10. Decrease the opacity until the lower greyscale 3D image emerges. Mine is at 50%.
11. The greyscale made the map dull. Open the Color - Brightness/Contrast. For the 2nd picture below, I just brought the contrast up by 40, which brings back the colors.

Now to check the map, I zoomed into the mountains in the central region of this map.

This is coming out very nicely, Notice the water of the ocean and the large river water on up to the highest snowcapped peaks.

I sharpen the map for my own eyes since this closeup looks kind of blurred to me.

From that plain color picture and the greyscale map, we have made a map that shows individual mountains, hills, valleys, together with shadows. The sun is coming from the northwest corner as set in the azimoth setting for this picture.

Here is the North Pole view using the same method.

The originals that were saved were GIF format. 

I found that the some maps need to be JPG. Some of the maps are GIF and need to be converted to JPG first. Otherwise the GIMP Bump function doesn't work.

And South Pole

A close up of the North Pole

A close up of the South Pole