So I’m pretty sure that this entry is only going to be interesting to me, but I guess that’s why this is MY blog!
I was craving a little mental exercise to do, so when I posted the Can’t Stop game a few weeks ago there were a plethora of things to start thinking about.
First, I wanted to know what the highest probability you could get would be and what the numbers were, so I made an Excel spreadsheet and found that 6, 7 & 8 are the best numbers and they give you a 92% chance of getting one of those 3 numbers with 4 die. What is the worst? If you need the numbers 2, 12 & (11 or 3) you only a 44% chance of getting one of those numbers with 4 die.
That was fun to figure out but I wanted something where I didn’t have to make a spreadsheet and crunch numbers out – I wanted a real formula. So, to get a real formula, I first had to come up with a real problem. The problem I came up with was to find the probability of rolling at least two 1′s with ‘n’ number of die.
Now, for 2 die, that’s pretty easy to figure out, it’s 1 chance in 6 for the first dice (has to be a 1) and 1 chance in 6 for the second dice (it also has to be a 1) for a total probability of 1 in 36 or 2.8%.
When we have 3 die, there gets to be a little kink in the solution… When you have a 1, 1, X (X is any number) or a 1, X, 1 or a X, 1, 1 – these are all valid so you would think that you have (6+6+6)/216 (where 216 is the total number of rolls that you can have with 3 die) for a total of 8.3% which is better than with 2 die but not quite correct because of ‘the kink’. The problem is that you’ve counted the combination 1, 1, 1 three times instead of just 1 time as it should be, so you should really have (6*3-2)/216 for a total of 7.4% which is correct.
The combinations that you start to count twice get a lot more complicated as you start adding more die to the mix giving the need for my precious formula which I found to be:

And, if I plug in the numbers for 3 die (n=3), I get the same answer – awesome!

I want to try this now for 4 die and see if it matches what I got from the manual method with Excel (result was 13.2%).

It does, one more check with 5 die and I’ve verified my results again (Excel said 19.6%).

With 6 die you’re really pushing the limits of manual calculation with Excel because I would need to have 46656 lines; it would be possible but also pretty time consuming. I plugged it into my formula to find that it gives me a realistic result of 26.3%.

If I wanted to check the probability with 7 die (it’s 33.0%), I would not be able to do that in Excel anymore as it can’t take a spreadsheet that size, so I would have to have a formula if I wanted to do that. Thank goodness I do!
For snake eyes, this wasn’t too complex to do; but if someone wanted to try it for rolling a 7 – that would get pretty hairy indeed. I can’t believe that you read this far – you’re such a geek too!