RPGs and Statistics
Funkysided dice. It's one of the fun and visual parts of RPGs that are easy to point out. From the gamer's caltrop D4 through to the golfball D100^{1}, they're how we can account for Random Probability of things happening to our beloved characters and carefully crafted plotlines alike. The Old Faithful uses the D20 as the main die of chance, and other games play with die pools, die sets and other ways to make random rolls feel different and result in different odds for/ against the character.
Roll, hit a target number
The system most people can grok quickly. Roll a die. Your result is the number rolled, plus or minus some other numbers depending on skill or situation. Big numbers are typically better.
The charts below are created from a script that performs the specified roll 10,000 times each time I run the program. Results are combined each time. Each roll has the result tabled; graphs are built from that table.
Dungeons and Dragons
The big D20. Grants a chance of a number between 1 and 20. Evenly. In combat, a roll of 1 is a botch and a roll of 20 is a critical (success). Looking at the chart, you've got a ... 1/20 chance of a 1 and a 1/20 chance of a 20. Indeed, a 1/20 (5%) chance of any number.
So, given a difficulty of 15 with no bonuses or penalties, you've got a 1/20 chance of rolling 20, 19, 18, 17, 16 or 15. That's six numbers, so a 6/20 or 3/10 (30%) chance. If you've got, say, a +1 then you also succeed on a 14. so that/s 7/20 (35%) chance. Each plus increases your chance by 5%, each negative decreases your chance by 5%.
Then they added what I feel is the greatest addition to the game in 5th Edition. Rolling with Advantage or Disadvantage. Roll 2 d20s  take the best for Advantage, or worst for Disadvantage.
With advantage, we've got a 1/20 chance of rolling a 20 on the first roll, and a 1/20 chance of rolling on the second. So... a 2/20 chance (or 1/10) of getting at least 1 20. Nice. For the lower value, we've got a 1/20 chance of rolling a 1 and a 1/20 chance of rolling another 1; required to get a botch in advantage. That's 1/20 * 1/20, which is 1/400 (0.25%) chance of botching with advantage. With disadvantage, the probabilities are flipped. More than just having a + or  to your roll, this changes the spread of probability.
But let's go back to the first difficulty of rolling a 15. For that to happen, we have to NOT roll 14 or less on both rolls. Which is 14/20 * 14/20 = 49/100: the inverse of that is 51/100. Just over half the time, roughly equivalent to having a +4 to the roll. Not bad, if you're taking the highest. If not... well... that's 3/10 * 3/10 = 9/100 (9%). Roughly, 4. Oof.
If we instead have the target of 11, that's 50% of the time normally, ~75% of the time at advantage (+5), and ~25% of the time at disadvantage (5).
FATE
FATE's die are funky, even among funky die. They're six siders, but two of them have + symbols, two are blank, and two have  symbols. They adjust your statistic score up or down depending on your roll, meaning your basic skill level is very important.
At the extreme ends we're talking 1/3 * 1/3 * 1/3 * 1/3 or 1/81 (~1%) of rolls. The next is better, as there are 4 possible combinations of [] [] [] [ ]. So 4/82 or 2/41 (~4%) of rolls. There are 19 different combations that net a score of 0, 19/81 is the obvious ratio there.
FATE's skill pyramid for starting characters goes from 15, higher being better. Average difficulty is 3, so for someone with using a '1' base skill, you're going to need at least +2, which happens X% of the time. With a '3' base skill, you need a +0 or better which is not 50%  it's actually 50/81 or 62%. Which means at an average task the average skilled fails 38% of the time. Not sure what that says about anything.
Powered by Apocalypse
Craps. In as much as you roll 2d6 and add the results together. You're more likely to get the middle result, much like FATE, but instead of it being banded around your stat; you add your stat to the roll. Which probabilitywise is kinda the same thing. However rather than having an increasing difficulty and an increasing skill (hopefully); you've always got the base fail/near success/success band.
Generally you've got a small band of + or  based on whatever statistics your flavour of PbtA favours. Maybe +2 in your best. So to get a nottotalfail you'll succeed just over 70% of the time. To full on succeed... well... with zero pluses that's about 17%. At +2, about 42%. I feel this sort of system supports continuedproblemsnothingevercompletelysolves. Because even with growth, you're going to fail frequently.
Savage Worlds
You could almost say that Savage Worlds pioneered the Advantage roll that DnD 5th gained. Characters in Savage Worlds have thier stats measured in die sides rather than points. So D4 is rubbish, D6 is average, then D8, D10 and finally D12... well, then you get D12+2 etc. but that's splitting hairs.
Basic difficulty target is a 4. Seems simple enough. On a D4 you've got a 1/4 (25%) chance of getting a 4+. On a D6, 1/2 (50%). D8 is 5/8 (62.5%). D10 is 7/10 (70%). D12 is 3/4 (75%).
Then we add an additional wrinkle: alongside the stat die, you as a player character roll a D6 to go with it. You get to pick whichever is highest for your total. Here I have to flip the math again  with a D4 you've got a 3/4 chance of not getting a 4; and add a D6 for 1/2 chance of not getting a 4 and you've got a 3/8 chance of not succeeding; so a 5/8 chance you will. Which you remember from above, is the D8 alone chance. To keep going, a 23/32 (71.875%) for D8, a 31/40 (77.5%) for D10 and 13/16 (81.25%) on D12.
AND THEN we add a final wrinkle: the die Explode. This means if you roll the maximum value on a die, you roll again and add that to your initial roll of that die. A roll that can also explode. Not only on your stat die, but on your additional D6 too. Boy does that play fun with probabilities. This is quite out of my depth, but I'll give it a crack at the D4 level.
You have a 1/4 chance of getting a 4. That explodes. You then have another 1/4 chance of getting a 4. That also explodes. So it's 1/4 * 1/4 * 1/4 etc. You then tag in a D6 with a 1/6 chance of exploding. Which has a 1/6 chance of exploding on another 6 etc. Which from the graphs below SUPER changes the result pattern. Just over a 60% chance of a 4 to ... just over 60% chance of a 4. However, a slim chance of a superpowered success. Which ensures superpowered successes are rare and exciting, but a character is quite likely to succeed basically at basic tasks.
Roll a pool, score successes
With this system you combine a number of die together to indicate your character's skill at a task. Roll them and tally the number of 'Successes'. That is your result, again higher is typically better.
The charts below are created from a script that performs the specified roll 10,000 times each time I run the program. Results are combined each time. Each roll has the resultant number of successes tabled; graphs are built from that table.
World of Darkness/ Chronicles of Darkness
This system has a pool comprised of 10 sided dice. In the original system, you scored a success for each die that had 6 or more for basic difficulty. Which equated to 1/2 the time (50% chance of success) per die. Difficulty could be changed up or down based on the situation (so a harder task might need 7 or up, or 2/5 (40% chance)). Later versions of the system kept the difficulty at 7 or 8, but changed the number of successes required (so a hard task might keep a difficulty of 7, but need 4 or more success). Base chance of rolling a number on a D10 didn't change.
However to spice things up the failure and success states are mechanised with Botch and Critical. If you roll a 1, it takes away one of your successes. If this leaves you with zero successes, you botch (so critical failure ala Dungeons and Dragons). If you roll a 10, you keep that success and reroll it to see if you get another. Other incarnations had for every two 10s you rolled, you scored 3 successes rather than 2. So for arguments sake, I've weighted rolls to be:
 1 is a 1 result (takes away a success)
 < difficulty gives 0 (no successes)

= difficulty gives to +1 (a success)
 10 gives 1 success and maybe more (reroll for 6 difficulty and 3 for 2 10s in the 8 difficulty charts below)
Creating a die pool of 4 for a starting character gives the following result spread
Shadowrun
Shadowrun uses d6s as their pooldie of choice, with a target number of 5. So at a baseline we're looking at 1/3 chance of success per die. Doing the reverse calculation from before, 2/3 * 2/3 chance of failure on 2 dice so 4/9, meaning 5/9 chance of success  immediately jumping up over half chance to succeed with one additional die. Add one die, 8/27 of fail or 19/27 chance of success. Converting the 2 die to denominator 27 gives it 12/27 vs 19/27. So a more modest jump. As we expect, the more dice added gives less boost, but still increases success.
Every system adds a wrinkle, Shadowrun has glitches  if more than half your pool land on 1 then you've glitched. So, it's possible with a big pool to succeed and glitch, which is neat. It does make my charts more complex.
 Black horizontal lines are Success chance, starting from the bottom
 Overlapping diagonals at the bottom show the Successes that were also Glitches
 Diagonals at the top of the horizontal lines are Failures that weren't glitches.
 Critical Glitches are vertical lines at the top of each bar.
I'm quite interested by the Critical Glitch chance bouncing based on odd/ even die numbers because of the morethanhalf pool.
Where I double checked
 D&D 5e: Probabilities for Advantage and Disadvantage
 Advantage and Disadvantage in D&D Next: The Math
 Fate Dice Statistics
Credits
 Dungeons and Dragons is © 19932020 Wizards of the Coast LLC, a subsidiary of Hasbro, Inc. All Rights Reserved. More on Dungeons and Dragons
 FATE is Copyright Evil Hat Productions. All Rights Reserved. More on FATE
 Powered by the Apocalypse is developed by Meguey Baker and Vincent Baker... and their trademark/ copyright status confuses me.
 Savage World is Copyright Great White Games LLC.
 World of Darkness/ Chronicles of Darkness are © White Wolf Entertainment AB  A Paradox Interactive AB company. All rights reserved. More on White Wolf
 The Topps Company, Inc. owns the Shadowrun intellectual properties. More on Shadowrun

A friend of mine has sourced me a D120. I have no idea what I'll use it for, and I don't have it yet as of the time of writing, but I already love it. ↩︎