Sample space, in full
The sample space, usually written as the Greek letter omega, is where a probability calculation starts. It lists what can happen, once each. A single coin flip has the two-element space of heads and tails. A d20 roll has the twenty elements 1 through 20. An event is any subset of that space, so rolling above 15 is the four-element subset 16, 17, 18, 19, 20. Getting the space right is most of the work, because a badly specified space produces confident wrong answers.
A 6-from-49 lottery draw has a sample space of 13,983,816 unordered six-number sets. The same physical draw could be described instead by its 10,068,347,520 ordered sequences, and both are legitimate spaces. The choice matters because prize rules ignore order, so the unordered space is the one that matches how the game actually pays. Powerball's space is larger again: 11,238,513 white-ball sets times 26 red balls, or 292,201,338 outcomes, which is the jackpot figure the operator publishes.
A sample space carries no probabilities of its own. Attaching them is a separate step, and the only requirements are that they are non-negative and sum to 1. For a fair d20 each of the twenty outcomes gets 1/20, which makes the distribution uniform. For the total of two d6 the space runs from 2 to 12, eleven outcomes, but they are not equally likely: 7 can be made six ways and 2 only one. Listing outcomes evenly does not make them even.
- Probability
- Outcome space, Possibility space
A worked example
Two six-sided dice. Describe the space as the 36 ordered pairs from (1,1) to (6,6) and each pair has probability 1/36, so the space is uniform. Describe it instead as the eleven possible totals, 2 through 12, and the outcomes are the same physical rolls but the probabilities are lopsided: 7 covers six of the pairs at 6/36, while 12 covers only (6,6) at 1/36.
Sources
Powerball's jackpot odds are 1 in 292,201,338, the size of the game's outcome space.
Powerball — Powerball Prize Chart
See also
Where it comes up on this site
Sample space: frequently asked
The whole numbers 1 to 20, one per face, each at probability 1/20 on a fair die.
Yes, and picking between them is a modelling decision rather than a fact about the drum of balls. A 6-from-49 draw can be written as 13,983,816 unordered sets or as 10,068,347,520 ordered sequences, and both are correct descriptions of the same event. Lotteries quote the unordered figure because prizes ignore the order the balls came out in, so that is the space the rules actually operate on.