Probability Describes a Chance; a Distribution Describes the Whole Bet

Probability measures the chance of an event. A probability distribution describes how probability is spread across the possible values of a random variable.

For a wager, that distinction matters. “How likely am I to win?” is only one question. To understand the bet’s price and risk, you also need to know how much each outcome wins or loses.

From a winning chance to a complete distribution

A random variable is a numerical result determined by chance. For gambling calculations, a useful choice is X = net profit or loss on one bet.

A discrete probability distribution lists the possible values of X and their probabilities. Each probability must be between 0 and 1, and the complete list must add to 1. The outcomes must not overlap or leave anything out. A table assigning a probability to each value represents a probability mass function, or PMF. Penn State’s probability lesson explains these definitions and properties.

Example: $10 on red in double-zero roulette

Assume:

• A fair wheel with 38 equally likely pockets: 18 red, 18 black, 0 and 00.
• Red pays 1:1: a winning $10 bet earns $10 profit, with the original stake also returned.
• Any non-red result loses the full $10; there is no half-loss or other special zero rule.

Island Resort & Casino’s rules describe the double-zero wheel, its red and black numbers, green zeros and forfeiture of losing wagers. The Venetian’s rules confirm the 1:1 payout on red or black.

The probability of winning is:

P(red) = 18 ÷ 38 ≈ 47.37%

But the distribution of net profit contains both possible results:

Result	Net profit, X	Probability
Red	+$10	18/38 ≈ 47.37%
Black, 0 or 00	−$10	20/38 ≈ 52.63%
Total		38/38 = 100%

There are 38 equally likely pocket outcomes, but only two possible money outcomes for this bet. Those money outcomes are not equally likely.

The distribution reveals expected loss and variance

Expected value is the probability-weighted average of the money outcomes. Variance measures their spread around that average; standard deviation is the square root of variance. These are calculated from the distribution, not from the win probability alone. Penn State gives the formulas.

For the $10 bet:

E(X) = (+$10 × 18/38) + (−$10 × 20/38) = −$20/38 ≈ −$0.5263

The expected loss is about 53 cents per $10 wagered. Dividing that loss by the stake gives a 5.26% house edge under these assumptions.

That average is not a possible result of one spin: you either gain $10 or lose $10.

Using variance = E(X²) − [E(X)]²:

Var(X) = (10² × 18/38) + ((−10)² × 20/38) − (−20/38)² ≈ 99.723 dollars²

Standard deviation = √99.723 ≈ $9.99

The spread is large compared with the expected loss. Standard deviation is not a loss limit or a guaranteed range; it describes variability. For repeated bets, the session probability model explains how expected loss and spread accumulate.

Which distribution do you need?

Discrete distributions suit countable results, such as net payouts or the number of wins. Continuous distributions describe values across an interval; probabilities are represented by areas under a density curve, not the curve’s height at one point. Penn State distinguishes these two types. A continuous model can sometimes approximate a large collection of discrete outcomes, but it is not the exact payout table.

Before evaluating a wager, define the money result, include every payout and loss—including a $0 net result for any push—and check that probabilities total 100%. Use net profit, not a winning return that includes your stake. Then calculate expected value and spread separately: the first prices the bet; the second explains why a short session can look very different from that price.

Neither calculation guarantees a winning session. Set an affordable loss limit before play, rather than treating an expected average as protection against losses.