When is it profitable to bet for the TrustDice Jackpot?

Contents

When betting at TrustDice (review, bettors are geo-blocked) you have the opportunity to win a jackpot on every roll you make. TrustDice put 7% of the profit they make from their dice game into the jackpot and so the jackpot prize is constantly increasing until it is won. The rules of the jackpot are:

The rules indicate 2 important facts about the TrustDice jackpot:

  1. The jackpot prize constantly increases until it is won
  2. The probability of winning the jackpot always remains the same

These 2 characteristics ensure that sooner or later the jackpot will grow large enough that betting on TrustDice is statistically profitable. We will now calculate how large the jackpot needs to be for this to be the case.

There are 2 methods we can use to find when the jackpot is so big that its profitable to bet for it:

  1. Estimate how many rolls (on average) are required to win the jackpot and then calculate how much it would cost to make that many rolls. If the jackpot is larger than this amount, then the jackpot is profitable.
  2. Use the formula for expected value to determine what the jackpot needs to be for the expected value to be greater than zero.

Method 1: Estimating the cost of rolling enough times to win

We first need to calculate the probability of winning the jackpot. To win the jackpot requires both of the following:

  1. You must win the dice roll
  2. Your jackpot roll must be “7777”

The combined probability of both of these outcomes is:

probability_jackpot = probability_dice_roll × probability_7777

Since we want to maximize our chance of winning the jackpot, we choose the highest possible “Roll Under” value of 96. This makes probability_dice_roll = 95%.

To calculate the probability of the jackpot roll being “7777” we observe the probability of any one digit in the jackpot roll being 7 is 10% or 0.1. The probability that all 4 digits are 7 is therefore 0.1 × 0.1 × 0.1 × 0.1. The probability of winning the jackpot is therefore:

probability_jackpot = 0.95 × 0.1 × 0.1 × 0.1 × 0.1 = 0.000095

This means the expected number of rolls to win the jackpot is:

rolls = 1 / probability_jackpot = 10526.316

When we select “roll under 96” we have a 95% chance of winning 1.0368 × bet_amount. So, on average, each bet will cost:

bet_cost = bet_amount - 0.95 × 1.0368 × bet_amount = 0.01504 × bet_amount

So the expected cost of rolling enough times to win the jackpot is:

cost = 10526.316 × 0.01504 × bet_amount = 158.316 × bet_amount

Winning the jackpot will return a prize of:

win = 0.8 × (bet_amount / max_bet_amount) × jackpot

We want to find the jackpot amount that makes win > cost. This means:

0.8 × (bet_amount / max_bet_amount) × jackpot > 158.316 × bet_amount

Dividing each side of the inequality by 0.8 × bet_amount / max_bet_amount yields:

jackpot > 158.316 × max_bet_amount / 0.8
jackpot > 197.895 × max_bet_amount

Putting in the max_bet_amount = 0.5 BTC we get:

jackpot > 98.947 BTC

So if the jackpot is 98.947 BTC or larger it is profitable to bet at TrustDice.

Method 2: Finding where expected value is positive

The formula for expected value is:

EV = PD × PJ × S × J + PD × W × B - B
PD : Probability that the dice roll wins
PJ : Probability that the jackpot roll wins
S : Share of the jackpot that the bettor will win
J : Size of the jackpot
W : Payout multiplier if the dice roll wins
B : Bet amount

Picking the highest probability of winning the dice roll makes PD = 0.95 and W = 1.0368. The jackpot share is 0.8 × B / Bmax. The probability of winning the jackpot roll is PJ = 0.14. Plugging these numbers into the formula for expected value gives:

EV = 0.95 × 0.1⁴ × 0.8 × B / Bmax × J + 0.95 × 1.0368 × B - B
   = 0.000076 × B / Bmax × J - 0.01504 × B

We want to find where EV > 0:

0.000076 × B / Bmax × J - 0.01504 × B > 0
0.000076 × B / Bmax × J > 0.01504 × B
J > 0.01504 × B / (0.000076 × B / Bmax)
J > 197.895 × Bmax

The TrustDice rules indicate Bmax = 0.5 BTC and so we have:

J > 98.947 BTC

So as long the jackpot is at least 98.947 BTC the expected value of betting is greater than zero and so betting is profitable.

Deciding how much to bet

Now we know how high the jackpot must be for betting to be profitable, the next question is how much to bet each roll. To qualify for the jackpot you must bet at least 0.001 BTC and the maximum allowed bet is 0.5 BTC so you will want to bet within this range. We can show that the bet amount doesn’t actually impact the return on investment:

ROI = EV / B × 100%
    = (0.000076 × B / Bmax × J - 0.01504 × B) / B × 100%
    = 0.0152 × J - 1.504

So if the jackpot is 100 BTC your expected ROI from a single bet is 0.8% whether you bet 0.001 BTC or 0.5 BTC or anything in between. However, assuming you are going to need to bet 10,526.316 times to win the jackpot, it will make a big difference in how much capital you will need and what total profit you can expect:

capital = 158.316 × B
profit = 0.8 × B / Bmax × J - 158.316 × B
return_on_capital = profit / capital × 100%

If the jackpot is 100 BTC we would have:

Bet per rollRequired capitalProfitReturn on capital
0.001 BTC0.158 BTC0.002 BTC1.64%
0.01 BTC1.583 BTC0.017 BTC1.64%
0.1 BTC15.832 BTC0.168 BTC1.64%
0.5 BTC79.158 BTC0.842 BTC1.64%

The profit in the table above assumes you stop betting after winning the jackpot but if you were to bet 0.001 BTC, hitting the jackpot would earn you 0.16 BTC and leave 99.84 BTC still in the jackpot. This means the jackpot is still profitable so you could keep betting and win the jackpot again and again. Under this model betting 0.001 BTC is no different than betting 0.5 BTC except you would need much less capital.

However, if we assume there are multiple people competing for the jackpot then perhaps you would want to bet big so when you win the jackpot you take a larger share of it before other players have a chance to win it.

Summary

It feels a bit disappointing that you can “win” a huge 100 BTC jackpot and you only get 0.16 BTC. This leaves 99.84% of the jackpot behind. Usually if it’s said you “won” something that does not mean you won 0.16% of it.

“I won a million dollar lotto!”

“Wow, are you going to buy a house?”

“Well I won $1,600″…

But this is the structure TrustDice has gone with and it give this jackpot hunt some interesting characteristics. Do compare this to our writeup of the Nitrogen Jackpot, which is sometimes +EV to bet for and our writeup of the chance of winning the Prime Dice jackpot.

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