When is it profitable to bet for the TrustDice Jackpot?

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 the 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 on 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: