# Mathematically profitable dice betting

Nitrogen (review) and FortuneJack’s (review) dice games have an accumulating jackpot which is won by rolling an unlikely series of numbers. Once the jackpot gets big enough it becomes profitable to play for it because the amount of the jackpot is more than the amount you expect to lose by playing for it.

## Positive expected value

In gambling expected value is the amount you can expect to win or lose from a bet on average given the mathematical probability of winning and the payout.

If you are betting on the flip of a coin where you win ฿2 if you win and you lose ฿1 if you lose you expect to win ฿2 half the time and lose ฿1 the other half of the time. So on average you expect to win ฿0.5 per flip.

You could still lose on an individual flip, you could still lose 10 flips in a row but it’s a good bet to make anyway because the odds are in your favor, you have an edge, the bet has positive expected value.

## Nitrogen’s jackpot

Nitrogen’s jackpot is won by rolling a number with a 7 in it 12 times in a row (sometimes it’s 13). In the long run you will make money by betting when the jackpot is over ฿3.65 because the chance of winning the jackpot is greater than the money you expect to lose trying to win it.

To work this out we need to know how many bets you need to make until you can expect to win the jackpot and how much money you expect to lose by making that many bets. So let’s look at the rules for winning the jackpot.

• Roll a number with a 7 in it 12 times in a row. The 7 can be any of the 4 numbers in your roll such as 70.00, 07.00, 00.70 or 00.07
• Each bet has to be for at least ฿0.001

You can see the full rules, terms and conditions here. As we figured out in this Bitcointalk Forum thread here is how we work out the chance of rolling a number with a 7 in it

``````
P(any number is 7) = 1 – P(no number is 7)
= 1 – P(first not 7) * P(second not 7) * P(third not 7 ) * P(forth not 7)
= 1 – 0.9 ^ 4
= 0.3439
``````

This tells us you have a 34.39% chance of rolling a number with a 7 in it. Next we work out how many times you have to roll to expect 12 consecutive numbers with a 7 in them

``````
0.3439^12 = 0.00000273644
1/0.00000273644 = 365438.155453
``````

This tells us you expect to hit the jackpot once in every 365,438 rolls.

The house has a 1% edge so for every ฿0.001 bet you expect to lose ฿0.00001. Next we multiply the amount of rolls times the expected cost per roll

``````
365438*0.00001 =  3.65438
``````

This tells us it costs ฿3.65438 to roll enough times that you can expect to hit the jackpot. As such any time the jackpot is more than ฿3.65438 you should be rolling until it is hit. The succinct calculation is

``````
1/((3439/10000)^12)*0.001*0.01 = 3.65
``````

Sometimes Nitrogen set the jackpot at 13 rolls or other numbers instead of 12 but all else remains equal so the calculation becomes

``````
1/((3439/10000)^13)*0.001*0.01 = 10.626
``````

So if the jackpot is set to require 13 rolls containing a 7 it will be profitable to bet after the jackpot reaches ฿10.626.

## How to best take advantage of this

The most optimal way to bet in this situation is to use Nitrogen’s auto roll betting bot that allows you to set the bet amount at ฿0.001 and roll continuously until someone hits the jackpot. If you find it more fun to bet manually or if you want to try different betting strategies then go for it but the more bets you make at ฿0.001 the better.

The settings indicated in the below screenshot of Nitrogen’s dice interface are highly recommended for optimal positive expectation betting.

On average you will only be losing 1% of turnover until you hit the jackpot, in fact you have a 49% chance of winning just from your rolls without the jackpot! You also get a free bet in Nitrogen’s excellent sportsbook every time you get halfway to the Jackpot.

If you sign up via our referral link you get entry into our exclusive 300 mBTC monthly poker freeroll. Let’s do it!

## FortuneJack’s jackpot

Another major player in the bitcoin dice market, FortuneJack also has a jackpot. As per the image above in order to win you have to roll a number that starts with a 1, then a 2 then a 3, then a 4, then a 5, then a 6 and then a 7. This should happen once every 10,000,000 (10 million) roles.

You then get a % of the jackpot based on how big your bets were. The full calculations for working out how big the jackpot has to be to make it profitable to play can be seen at this Bitcointalk Forum thread.

If you are betting ฿0.000001 the jackpot needs to be over ฿0.5 to be +EV.
If you are betting ฿0.0001 the jackpot needs to be over ฿20 to be +EV.
If you are betting ฿0.01 the jackpot needs to be over ฿1250 to be +EV.

FortuneJack also have an auto betting bot to facilitate optimal betting for this jackpot.

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### 5 Comments

1. Vlol 11 months ago

hi, how many EV i get for every 1 satoshi over 10.62642 BTC? I mean: how can i calculate how profitable my every spin in this game depends on jack pot size?

• Bitroll 11 months ago

Great question Vlol! I think your expected value is the whatever the jackpot is as a percentage of the break even point for the jackpot.

Needing 13 sevens the break even point is 10.62642

If the jackpot is 11.689062 that is 10% more than the break even point so you have a 10% positive expected value. That means for every 0.001 you bet you can expect to return 0.0011!

I am not totally sure about that so I would love a maths whizz to confirm or deny it.

• Robbie 11 months ago

Yep, you’re math checks out:

EV = P(Win) * Jackpot – P(Lose) * Bet = P(Win) * Jackpot – [1 – P(Win)] * Bet

a bit of algebra and you can show

P(Win) = Bet / (BreakEvenJackpot – Bet)

and therefore

EV / Bet = [(Jackpot – Bet) / (BreakEvenJackpot – Bet)] – 1

if your bet is small compared to the jackpot, you can approximate this as

EV / Bet = Jackpot / BreakEvenJackpot – 1

so if Jackpot = BreakEvenJackpot + 10% your EV = 10% of your bet!

• Anonymous 11 months ago

now there is jackpot 13.7 BTC. winner will take all this price? jack pot will be equal to 0 after that? Seams like it is very profitable play now, why there is no rush on this site? why you guys do not playing now? about 23 cent of EV?

• Bitroll 11 months ago

Yes the winner takes the whole jackpot. After that the jackpot does not rest to 0 but the funds for the new jackpot do not get directly taken out of the previous jackpot winners winnings. I have been playing!