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tly you do that by computing something called standard deviation and the larger that is the more random it is but that takes more computation than i want to get into today i ll get into it later on all i m asking you to know is that in general fewer dice rolled more random while i m on the subject more faces on a die is also more random since you have a wider spread computing probability through counting you might be wondering how can we figure out the exact probability of getting a specific roll this is actually pretty important in a lot of games because if you re making a die roll in the first place there is probably some kind of optimal result and the answer is we count two things first count the total number of ways to roll dice no matter what the result then count the number of ways to roll dice that get the result you actually want divide the first number by the second and you ve got your probability multiply by 100 if you want the percentage examples here s a very simple example you want to roll 4 or more on 1d6 there are 6 total possible results 1 2 3 4 5 or 6 of those 3 of the results 4 5 or 6 are a success so your probability is 3 divided by 6 or 0 5 or 50 here s a slightly more complicated example you want to roll an even number on 2d6 there are 36 total results 6 for each die and since neither die is influenced by the other you multiply 6 results by 6 to get 36 the tricky thing with questions like this is that it s easy to double count for example there are actually two ways to roll the number 3 on 2d6 1 2 and 2 1 those look the same but the difference is which number appears on the first die and which appears on the second die if it helps think of each die as having a different color so maybe you have a red die and a blue die in this case and then you can count the ways to roll an even number 2 1 1 4 1 3 4 2 2 4 3 1 6 1 5 6 2 4 6 3 3 6 4 2 6 5 1 8 2 6 8 3 5 8 4 4 8 5 3 8 6 2 10 4 6 10 5 5 10 6 4 12 6 6 it turns out there are exactly 18 ways to do this out of 36 also 0 5 or 50 perhaps unexpected but kind of neat monte carlo simulations what if you have too many dice to count this way for example say you want to know the odds of getting a combined total of 15 or more on a roll of 8d6 there are a lot of different individual results of eight dice so just counting by hand takes too long even if we find some tricks to group different sets of rolls together it still takes a really long time in this case the easiest way to do it is to stop doing math and start using a computer and there are two ways to do this the first way gets you an exact answer but takes a little bit of programming or scripting you basically have the computer run through every possibility inside a for loop evaluating and counting up all the iterations total and also all the iterations that are a success and then have it spit out the answers at the end your code might look something like this int wincount 0 totalcount 0 for int i 1 i 6 i for int j 1 j 6 j for int k 1 k 6 k insert more loops here if i j k 15 wincount totalcount float probability wincount totalcount if you don t know programming but you just need a ballpark answer and not an exact one you can simulate it in excel by having it roll 8d6 a few thousand times and take the results to roll 1d6 in excel use this formula floor rand 6 1 when you don t know the answer so you just try it a lot there s a name for that monte carlo simulation and it s a great thing to fall back on when you re trying to do probability calculations and you find yourself in over your head the great thing about this is we don t have to know the math behind why it works and yet we know the answer will be pretty good because like we learned before the more times you do something the more it tends towards the average combining independent trials if you re asking for several repeated but independent trials so the result of one roll doesn t affect other rolls we have an extra trick we can use that makes things a little easier how do you tell the difference between something that s dependent and something that s independent basically if you can isolate each individual die roll or series as a separate event then it is independent for example rolling a total of 15 on 8d6 is not something that can be split up into several independent rolls since you re summing the dice together what you get on one die affects the required results of the others because it s only all of them added together to give you a single result here s an example of independent rolls say you have a dice game where you roll a series of d6s on the first roll you have to get 2 or higher to stay in the game on the second roll you have to get 3 or higher third roll requires 4 or higher fourth roll is 5 or higher and fifth roll requires a 6 if you make all five rolls successfully you win here the rolls are independent yes if you fail one roll it affects the outcome of the entire dice game but each individual roll is not influenced by the others for example if you roll really well on your second roll that doesn t make you any more or less likely to succeed on future rolls because of this we can consider the probability of each roll separately when you have separate independent probabilities and you want to know what is the probability that all of them will happen you take each of the individual probabilities and multiply them together another way to think about this if you use the word and to describe several conditions as in what is the probability that some random event happens and that some other independent random event happens figure out the individual probabilities and multiply no matter what you do do not ever add independent probabilities together this is a common mistake to see why it doesn t work imagine a 50 50 coin flip and you re wondering what is the probability that you ll get heads twice in a row the probability of each is 50 so if you add those together you d expect a 100 chance of getting heads but we know that s not true because you could get tails twice if instead you multiply you get 50 50 25 which is the correct probability of getting heads twice example let s go back to our d6 game where you have to roll higher than 2 then higher than 3 and so on up to 6 in this series of 5 rolls what are the chances you make all of them as we said before these are independent trials so we just compute the odds of each roll and then multiply together the first roll succeeds 5 6 times the second roll 4 6 the third 3 6 the fourth 2 6 and the fifth roll 1 6 multiplying these together we get about 1 5 so winning this game is pretty rare so you d want a pretty big jackpot if you were putting that in your game negation here s another useful trick sometimes it s hard to compute a probability directly but you can figure out the chance that the thing won t happen much easier here s an example suppose we make another game where you roll 6d6 and you win if you roll at least one 6 what are your chances of winning there are a lot of things to compute here you might roll a single 6 which means one of the dice is showing 6 and the others are all showing 1 5 and there are 6 different ways to choose which die is showing 6 and then you might roll two 6s or three or more and each of those is a separate computation and it gets out of hand pretty quickly however there s another way to look at this by turning it around you lose if none of the dice are showing 6 here we have six independent trials each of which has a probability of 5 6 the die can roll anything except 6 multiply those together and you get about 33 so you have about a 1 in 3 chance of losing turning it around again that means a 67 or 2 in 3 chance of winning the most obvious lesson here is that if you take a probability and negate it just subtract from 100 if the odds of winning are 67 the odds of not winning are 100 minus 67 or 33 and vice versa so if you can t figure out one thing but it s easy to figure out the opposite figure out that opposite and then subtract from 100 combining conditions within a single independent trial a little while ago i said you should never add probabilities together when you re doing independent trials are there any cases where you can add probabilities together yes in one special situation when you are trying to find the probability for several non overlapping success criteria in a single trial add them together for example the probability of rolling a 4 5 or 6 on 1d6 is equal to the probability of rolling 4 plus the probability of rolling 5 plus the probability of rolling 6 another way of thinking about this when you use the word or in your probability as in what is the probability that you will get a certain result or a different result from a single random event figure out the individual probabilities and add them together one important trait here is that you add up all possible outcomes for a game the combined probabilities should add up to exactly 100 if they don t you ve done your math wrong so this is a good reality check to make sure you didn t miss anything for example if you were analyzing the probability of getting all of the hands in poker if you add them all up you should get exactly 100 or at least really close if you re using a calculator you might get a very slight rounding error but if you re doing exact numbers by hand it should be exact if you don t it means there are probably some hands that you haven t considered or that you got the probabilities of some hands wrong so you need to go back and check your figures uneven probabilities so far we ve been assuming that every single side on a die comes up equally often because that s how dice are supposed to work but occasionally you end up with a situation where there are different outcomes that have different chances of coming up for example there s this spinner in one of the nuclear war card game expansions that modifies the result of a missile launch most of the time it just does normal damage plus or minus a few but occasionally it does double or triple damage or blows up on the launchpad and damages you or whatever unlike the spinner in chutes ladders or a game of life the nuclear war spinner has results that are not equally probable some results have very large sections where the spinner can land so they happen more often while other results are tiny slivers that you only land on rarely now at first glance this is sort of like that 1 1 1 2 2 3 die we were talking about earlier which was sort of like a weighted 1d3 so all we have to do is divide all these sections evenly find the smallest unit that everything is a multiple of and then make this into a d522 roll or whatever with multiple sides of the die showing the same thing for the more common results and that s one way to do it and that would technically work but there s an easier way let s go back to our original single standard d6 roll for a normal die we said to add up all of the sides and then divide by the number of sides but what are we really doing there we could say this another way for a 6 sided die each side has exactly 1 6 chance of being rolled so we multiply each side s result by the probability of that result 1 6 for each side in this case then add all of these together doing this we get 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6 which gives us the same result 3 5 as we got before and really that s what we re doing the whole time multiplying each outcome by the probability of that outcome can we do this with the nuclear war spinner sure we can and this will give us the average result if we add these all together all we have to do is figure out the probability of each spin and multiply by the result another example this technique of computing expected value by multiplying each result by its individual probability also works if the results are equally probable but weighted differently like if you re rolling dice but you win more on some rolls than others as an example here s a game you might be able to find in some casinos you place a wager and roll 2d6 if you roll the lowest three numbers 2 3 or 4 or the highest four numbers 9 10 11 or 12 you win an amount equal to your wager the extreme ends are special if you roll 2 or 12 you win double your wager if you roll anything else 5 6 7 or 8 you lose your wager this is a pretty simple game but what is the chance of winning we can start by figuring out how many times you win there are 36 ways to roll 2d6 total how many of these are winning rolls there s 1 way to roll two and 1 way to roll twelve there are 2 ways to roll three and eleven there are 3 ways to roll four and 3 more ways to roll ten there are 4 ways to roll nine adding these all up there are 16 winning rolls out of 36 so under normal conditions you win 16 times out of 36 slightly less than 50 ah but two of those times you win twice as much so that s like winning twice so if you play this game 36 times with a wager of 1 each time and roll each possible roll exactly once you ll win 18 total you actually win 16 times but two of those times it counts as two wins since you play 36 times and win 18 does that mean these are actually even odds not so fast if you count up the number of times you lose there are 20 ways to lose not 18 so if you play 36 times for 1 each you ll win a total of 18 from the times when you win but you ll lose 20 from the twenty times you lose as a result you come out very slightly behind you lose 2 net on average for every 36 plays you could also say that on average you lose 1 18 of a dollar per play you can see how easy it is to make one misstep and get the wrong probability here permutations so far all of our die rolls assume that order doesn t matter rolling a 2 4 is the same as rolling a 4 2 in most cases we just manually count the number of different ways to do something but sometimes that s impractical and we d like a math formula here s an example problem from a dice game called farkle you start each round by rolling 6d6 if you re lucky enough to roll one of each result 1 2 3 4 5 6 a straight you get a huge score bonus what s the probability that will happen there are a lot of different ways to have one of each the answer is to look at it this way one of the dice and only one has to be showing 1 how many ways are there to do that six there are 6 dice and any of them can show the 1 choose that and put it aside now one of the remaining dice has to show 2 there s five ways to do that choose it and put it aside continuing along these lines four remaining dice can show 3 three dice of the remaining ones after that can show 4 two of the remaining dice after that can show 5 and at the end you re left with a single die that must show 6 no choice involved in that last one to figure out how many ways there are to roll a straight we multiply all the different independent choices 6x5x4x3x2x1 720 that seems like a lot of ways to roll a straight to get the probability of rolling a straight we have to divide 720 by the number of w...