Probability Modelling: From One Bet to a Session

Probability modelling means assigning probabilities to possible outcomes, then using those probabilities to answer a specific question. For a wager, that might be: What is its average cost? How widely could results vary? What is the chance of finishing ahead?

These are different questions. Expected value answers the first; a distribution of possible results helps answer the others. Neither predicts your next outcome.

Start with the rules, not recent results

Consider a hypothetical $10 bet on red in single-zero roulette. The model assumes:

37 pockets: 18 red, 18 black and one green zero.
Every pocket is equally likely.
Red pays exactly 1:1: a win returns the stake plus $10 profit.
Black or zero loses the entire $10 stake; no special zero protection applies.
Successive spins are independent.

The Maryland Lottery and Gaming Control Agency’s roulette rules document this wheel layout, a minimum red payout of 1:1, and loss on zero. This example uses exactly 1:1. Equal probabilities and independence are explicit assumptions of the model, not conclusions drawn from a short run of spins.

Group the pockets by their effect on the bet:

Outcome	Probability	Net result
Red	18/37	+$10
Black or zero	19/37	−$10

The probabilities add to one, and the results include the lost stake. Those are two basic checks for any wager model. A push, if the rules allowed one, would need its own row with a net result of $0.

Calculate the long-run price

Multiply each net result by its probability and add:

EV = (18/37 × $10) + (19/37 × −$10) = −$10/37 ≈ −$0.2703.

The expected loss is about 27 cents per $10 wager. Dividing that cost by the stake gives a 2.70% house edge under these assumptions. Our expected value guide explains this calculation in more detail.

Notice that the probability of winning is about 48.65%, not 50%. “Even money” describes the payout, not the chance of success.

Extend the model to 100 bets

Now suppose the session consists of exactly 100 independent spins, with $10 on red each time. There are no stake changes or early stops, and enough money is available to complete every bet.

Let W be the number of wins. It follows a binomial distribution because there is a fixed number of independent trials, each with the same win probability and two wager outcomes. These are the binomial conditions set out by Penn State.

For exactly k wins:

P(W = k) = C(100, k) × (18/37)^k × (19/37)^(100−k)

Here, C(100, k) counts the ways those wins can be arranged among 100 spins.

Convert wins into money:

Session net result = $10W − $10(100 − W) = $20W − $1,000.

So 45 wins means a $100 loss; 55 wins means a $100 profit. Finishing ahead requires at least 51 wins. Its probability is the sum of the formula above for k = 51 through 100. In Excel, calculate it with:

=1-BINOM.DIST(50,100,18/37,TRUE)

With TRUE, BINOM.DIST returns the probability of at most 50 wins; subtracting that from one gives the chance of finishing ahead.

The expected session result is simpler:

100 × (−$10/37) ≈ −$27.03.

That is a long-run average across repeated sessions, not a forecast or a loss limit.

Put variance beside expected loss

For a binomial win count, variance is np(1−p), and standard deviation is its square root, as Penn State’s derivation explains. Here, n is 100 bets and p is 18/37.

The variance of W is:

100 × (18/37) × (19/37) ≈ 24.98.

Because each extra win changes the session result by $20, the standard deviation in money is:

$20 × √24.98 ≈ $99.96.

That spread is much larger than the $27.03 expected loss. A profitable session is therefore compatible with a negative-EV bet. Standard deviation measures spread; it does not cap losses or define a guaranteed range.

Know when to change the model

An extra zero, a different payout or a refund on zero changes the outcome table. Changing stakes changes the money attached to each outcome. Stopping at a loss limit changes the session distribution, so the fixed-100-bet calculation no longer describes it.

Simulation can help explore more complicated stopping rules, but it still needs correct probabilities, payouts and bankroll constraints. Simulating this negative-EV bet does not change its expected return.

Use the model to understand cost and uncertainty—not to justify chasing losses. If you gamble, set an affordable spending limit before play; expected loss is not a safe bankroll allowance.