I recently passed my Statistics class in my game design degree, so I decided to fire up Excel to calculate Double or Nothing’s best probable statistics so I can push buttons in the most efficient way possible. Here are my results:
Explanation:
This chart represent the statistical game form of “everything that can happen in the game”.
0x Is actually the decision not to play the game, an actual possibility. You gain nothing, but it happens 100% of the time if you decide to.
For 1x, since you win half the time, and winning gives you 25 Units, your Net Gain Proportion is essentially half of your gain. I rounded it all down assuming Units aren’t awarded rounded up in these equations.
This actually introduces a bell curve up to 1.96x or 24 Units of your gains. The recommended strategy is stopping at 3x or 4x since that’s effectively close to this curve to double your total winnings.
In plain English, if you attempt to always double until you hit 20x, then ALWAYS cash that out, you will get 26,214,375 Units. However, once you spend 13,107,187 Units attempting it, you should have gotten it statistically speaking.
In short, I proven that anyone that goes above 5x is a dummy. 
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This is how you get banned from Casinos
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They would, but normally casinos don’t give net profits to people, or else they’d be bankrupt. TU’s Casino purposely does give us a bone in this case because contrary to popular belief, pixeltailgames is not siphoning all of your units to fund the architect in charge of the monorail construction.
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In all seriousness, this is pretty interesting to look over. Thanks for doing the math on this! 
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I’ll use this thread for any more Casino stats. Double or Nothing is probably the most interesting in this aspect. Some other tidbits I actually noticed just from watching some other machines.
As said, it is 100% worthless to go above 5x. To elaborate further on this, the net gain created by this probability curve goes on endlessly to 2 times the original base cost of going in. The proportional net gain is actually simply given by multiplying the net gain by the chance of winning, which is what is used for actual real-world “Game Expected Values” and apparently what smart people use to know their odds at winning raffles.
Video Blackjack pays out 5x your input, and you have somewhat of a 50/50 on that one, making it better than Double or Nothing already statistically, since 50/50 on Double or Nothing is 2x.
To actually determine “the best machine for making money” there are two primary variables. How fast you can feed the machine money itself, and the net gain. To actually be faster than something like Video Blackjack, the 3 main machines probably need a net gain of 30x or something, due to how slow it is to feed units to it. It actually doesn’t matter much else, since you’re always statistically going to net gain whatever you put in.
Double or Nothing was the easiest machine to throw in here. Other machines require me to actually tally how many of each result I get, as well as introducing the jackpot mechanics as something that can also happen with every probability. This does allow me to get pretty close to the “chance of getting something” but the roulettes kinda know when it’s about to get a “3 in a row” for specific stuff, so this will take a long time in general cause I also have to tally successes for each result and to get something accurate will probably take a ton of runs. Unlike Double or Nothing, the roulettes don’t give the control to the player in the payout.
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Actually, the chance of getting a double is not 50%. I made a post with a tally of how often you get a double. Wouldn’t it be the case that if the probability of getting a double is 50% there is no strategic way of playing the game? You have a 50% chance of doubling and a 50% chance of getting nothing, so the net is 0. What is P Net Gain and where did you get the multiplier?
If you flip a coin twice and it lands on heads both times, does that mean the chance of getting heads is 100%. No, of course not. The chance is 50%. You just got lucky.