Bonus analysis formula sheet

Define the measurement point, copy the exact terms, and list every outcome before using a shortcut. Include taxes, fees, expiry, caps, voids, and rule failures when they change cash flows. Results are estimates under stated assumptions, not guarantees.

The expected value chapter develops the core method. The bonus calculator can organize scenarios.

Odds and implied probability

Let D be decimal odds and A be American odds.

If A is positive: D = 1 + (A / 100)
If A is negative: D = 1 + (100 / absolute value of A)

If D is 2.00 or higher: A = (D - 1) x 100
If D is below 2.00: A = -100 / (D - 1)

Implied probability = 1 / D

Decimal odds include the returned cash stake. Positive American odds show profit on $100; negative odds show the stake needed for $100 profit. Implied probability is not necessarily true probability.

Cash-wager outcomes and expected value

Let S be a cash stake, p the estimated true win probability, and D the decimal odds. Assume two outcomes, no push, no fee, and full acceptance.

Winning total return = S x D
Winning net outcome W = S x (D - 1)
Losing net outcome = -S

EV = (p x W) + ((1 - p) x -S)
EV = S x ((p x D) - 1)

For any complete set of outcomes, let p_i be an outcome’s probability and X_i its net cash result:

EV = sum of (p_i x X_i)

Probabilities must total 1. Returned stakes are not profit. See free spins and insured bets for promotional payoffs.

ROI and break-even probability

Let C be the cash capital at risk for the measured decision.

Expected ROI = EV / C

A token already issued may have no cash stake, so dividing by zero is meaningless. For winning profit W and losing amount L:

Break-even probability = L / (W + L)

For an ordinary cash wager, L = S, so this simplifies to:

Break-even probability = 1 / D

A refund, boost, or cap requires revised net outcomes.

Wagering and contribution

Let B be the base defined by the offer and M its wagering multiple.

Published wagering target T = B x M

B is the exact base, such as bonus only or deposit plus bonus.

For handle block i, let H_i be actual eligible stakes and c_i the contribution rate:

Credited progress P = sum of (H_i x c_i)

If all planned handle has one contribution rate c:

Contribution-adjusted handle needed H = remaining target / c

Let each w_i be a planned handle share, with all shares summing to 1. Weighted contribution is sum of (w_i x c_i). Divide the target by that rate. Actual mix controls progress. See wagering requirements and game weighting and contributions.

Theoretical loss and ending balance

For casino handle block i, let r_i be modeled RTP and e_i = 1 - r_i be house edge:

Theoretical loss TL = sum of (H_i x e_i)

For one constant block:

TL = eligible handle x (1 - RTP)

Theoretical loss is an average, not a session prediction. Assuming completion, survival, full conversion, and no cap:

Simple expected ending balance = deposit + bonus amount entered - TL

If only part of the displayed bonus can become cash, replace the entered bonus with that lower convertible amount.

For a general sequence, let B0 be starting bankroll and EV_j each decision’s expected net result after all modeled costs:

Expected ending bankroll = B0 + sum of EV_j

Dependence does not prevent adding expectations, but it changes drawdown risk. See bankroll management basics.

Cashback

Let E be eligible net loss under the offer, r the rebate rate, K the rebate cap, and c the expected cash conversion rate of the credited form.

Gross rebate = E x r
Credited rebate = minimum(gross rebate, K)
Cash-equivalent rebate = credited rebate x c
Revised losing outcome = original losing outcome + cash-equivalent rebate

Set c = 1 only for withdrawable cash. Apply tiers, netting, and caps to each branch. See cashback and rebates.

Two-outcome cash hedge

Let S_A be cash staked on A at D_A and H the cash hedge on B at D_B. Assume exactly two outcomes, matched rules, full acceptance, and no fees, voids, limits, or price changes.

H = (S_A x D_A) / D_B
Equal gross return R = S_A x D_A
Equalized net cash result = R - S_A - H

Rounding can leave a small difference. A draw or rule mismatch breaks the model.

Stake-not-returned bonus-bet hedge

Let F be stake-not-returned bonus face value on A at D_A, and H a cash hedge on B at D_B. Use the same assumptions.

Bonus-bet winning cash K = F x (D_A - 1)
H = K / D_B

If A wins: converted cash = K - H
If B wins: converted cash = H x (D_B - 1)

The amounts equalize before rounding. Cash liquidity is H. This fails if the stake returns or winnings stay restricted. See matched betting fundamentals and confirm hedging is permitted.

Conversion rate

Let V be expected or equalized withdrawable cash from a promotional credit and F its face value:

Conversion rate = V / F

Label it expected or equalized. It is not full-promotion ROI.

Binary-wager variance and standard deviation

Let a fixed wager produce W on a win and -L on a loss, with win probability p. Assume exactly two outcomes and fixed payoffs.

Mean mu = (p x W) + ((1 - p) x -L)

Variance = p x (W - mu)^2
         + (1 - p) x (-L - mu)^2

Standard deviation = square root of variance

Variance uses squared currency units; standard deviation uses currency units. For n independent identical wagers, sum variance is n x one-wager variance and sum standard deviation is (square root of n) x one-wager standard deviation.

Kelly fraction

Let b = D - 1, p be estimated true win probability, and q = 1 - p.

Full Kelly fraction f = ((b x p) - q) / b
Fractional Kelly stake = bankroll x f x chosen fraction

If f is zero or negative, the formula calls for no wager. Kelly assumes reliable probabilities and payoffs. Model error and personal limits can require less or no wager. It does not make gambling safe.