Ok so I was playing double or nothing and I noticed that it wasn’t actually a 50% chance to double, but felt like it was a bit more in the players favor, so I made a tally of when I get a double and when I get nothing and here is what I got after 1200 tries:
Double: 707
Nothing: 493
It calculates to a 59% chance that a player doubles and 41% that a player gets nothing, so it’s likely that the devs decided on a 60% chance that you would double. Now what would be the strategy to earn the most amount of money in this case? Here is a formula I wrote that would express the net gain:
Net gain = un*(2^x)*(p^x) - un
where u=units spent on an attempt n=number of attempts p=probability of a double x=number of doubles per attempt
We already estimated p=0.6 and let u and n = 1 to simplify the equation you get:
Net gain = 1.2^x - 1
From this you can see that the more you double per attempt, the higher the gain, so it is always in your advantage to double.
I attached a picture of table showing how much you would have to spend (on average) to get different double multipliers and how much you would gain. For example, if you spend more than 88,618 units on double or nothing (without cashing out), there is more than a 50% chance that you would get a x16 Double (1,638,400 units)
EDIT
I would just like to point out that I made the assumption that the probability of getting a double is constant. This is likely not be true. I did notice that getting a double gets harder as you double more. I tallied the number of doubles I got for each double multiplier, but getting the data for higher doubles is difficult and time consuming and I do need more data to be confident. Here is what I got:
Attempts 1000 | Net units if I cash out here each attempt
x1 Double 585 | 4250 Units
x2 Double 339 | 8900 Units
x3 Double 182 | 11400 Units
x4 Double 104 | 16600 Units
x5 Double 54 | 18200 Units
x6 Double 33 | 27800 Units
x7 Double 13 | 16600 Units
x8 Double 8 | 26200 Units
x9 Double 4 | 26200 Units
x10 Double 0 | -25000 Units
Notice I don’t have much data for the higher doubles, so it’s not accurate for higher doubles, but you can see that the net units increase until a certain point (may be around x6?). It makes sense why the devs would make probability of a double lower for higher doubles, because it can be abused to get a ridiculous amount of money.
