# Your chance of winning the PrimeDice jackpot

### Contents

When betting on Primedice you have the opportunity to win a jackpot by rolling the number 77.77 twice in a row. The jackpot prize is an amount of bitcoin which steadily increases until the jackpot is won. When the jackpot is not won for some time the prize can get very large, to the point where betting on Primedice is statistically expected to be profitable.

Below we calculate

• Your chance of winning the jackpot.
• How high the jackpot needs to be for positive expected value betting?
• Optimal betting strategy to profit off the jackpot.

## What’s the probability of hitting the jackpot?

There are 10,000 possible numbers you can roll from 00.00 to 99.99.

77.77 is only 1 of those so you have a 1 in 10,000 chance of rolling it, which is a 0.0001% chance. You have to multiply that to roll it twice, so you get

``0.0001 * 0.0001 = 0.00000001``

or another way to put it, the number of expected rolls.

``10,000 x 10,000 = 100,000,000``

That’s a 1 in 100,000,000 chance!

## When does the jackpot make Primedice betting profitable?

TLDR; It needs to be over 10,000฿ to be profitable to bet for, which will never happen.

Oh well, we have better news about the Nitrogen dice jackpot which does sometimes get so big you can bet for it profitably, and you have a better chance of winning it, as we detailed in our mathematically profitable dice betting blog.

The Primedice jackpot becomes profitable when the amount you will win from hitting the jackpot exceeds the expected losses from the rolls required to hit the jackpot. The share of the jackpot you will win is either 1%, 10% or 100%, depending on the size of your bet. Your expected losses from each dice roll are the size of your bet multiplied by Primedice’s edge of 1%. So the jackpot is profitable when:

``jackpot × share > expected_rolls_to_hit_jackpot × bet × 0.01``

The share of the jackpot you will win is:

It doesn’t make sense to bet more than the minimum required to win a particular share of the jackpot since it will only increase your expected loss per roll. So you should only consider betting 0.0001, 0.001 or 0.01 ฿. We can see that in each of these cases:

``share = 100 × bet``

so our equation above becomes:

``jackpot × 100 × bet > expected_rolls_to_hit_jackpot × bet × 0.01``

We can divide both sides by `100 × bet` and get:

``jackpot > expected_rolls_to_hit_jackpot × 0.0001``

Whichever of these 3 bet sizes you use your expected value is the same.

So now we only need to calculate the expected number of rolls to win the jackpot. If we write out a sequence of dice rolls as `H` if we hit the jackpot number (77.77) and `T` if we miss it, it will look something like this:

``TTTTTTTTTHTTTTTTHTTHTTTTTTTTTHT...TTHTTTTTTTHH``

The final `HH` indicates we rolled 77.77 twice in a row and so won the jackpot. This looks exactly like a sequence of flips of a weighted coin where the probability of landing on heads is the probability of rolling 77.77. So the number of rolls before we hit the jackpot is the same as the expected number of flips of a weighted coin before it lands on heads twice in a row. This can be shown to be:

``expected_rolls_to_hit_jackpot = (1 + probability_of_7777) / probability_of_7777²``

There are 10,000 possible numbers that can come up each roll and only 1 of them is 77.77 so we have

``probability_of_7777 = 1/10,000 = 0.0001``

and so

``expected_rolls_to_hit_jackpot = (1 + 0.0001) / (0.0001 × 0.0001) = 100,010,000``

So on average, it will take just over 100,000,000 rolls to win the jackpot. For the jackpot to be profitable then, it must be worth at least:

``jackpot > expected_rolls_to_hit_jackpot × 0.0001 = 100,010,000 × 0.0001 = 10,001฿``

Yes, that’s over 10,000 bitcoin or about 320,000,000 USD.

## How to bet on the Primedice Jackpot

If the jackpot ever exceeded 10,001 ฿ (which it won’t) and you decide to take advantage of the expected positive return, you have 2 decisions to make:

• How much to bet each roll
• What “win chance” to set for each roll

Neither of these changes your expected value but we still have to choose something so let’s consider them.

When it comes to how much to bet you should only consider 0.0001, 0.001 or 0.01 ฿ for each roll. Each of these will result in a different variance of returns. The variance is how much you would expect your profit to vary if you were to replay the same betting scenario multiple times. Low variance means your profit would always be about the same whereas high variance means you would expect to make a loss some of the time and a large profit other times.

In the case of the Primedice jackpot, the more bets you are able to place the lower your variance. This means if you don’t have an infinite amount of bitcoin to bet with and you prefer something closer to a sure thing then you would be better off making a large number of small bets than a smaller number of large bets and so you should bet 0.0001 ฿ each roll.

The 2nd decision you need to make is the “win chance”. You can choose a low win chance which means that fewer of your rolls will win but you will get a bigger return when they do, or a higher win chance which means you will win more often but your return will be less. This choice potentially impacts the number of bets you can afford to make because if you set the win chance too low you increase the risk of running out of funds due to a losing streak. If you want to play it safe and give yourself the most chance to win the jackpot you should set the win chance to the highest allowed value: 98.

Either way, if you have an epic winning streak and find yourself in profit from the rolls without having hit the jackpot take your profits and stop trying to hit the jackpot unless you are having loads of fun.

These are the settings we recommend:

• Bet Amount: 0.0001 ฿
• Number of Bets: The highest number allowed is 999,999,999, which is almost as many as we should need!
• Win Chance: 98
• On Win: Reset
• On Loss: Reset
• Stop on Profit: 0.0 (disabled)
• Stop on Loss: 0.0 (disabled)
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