Sample Spaces & Events

The mathematics of uncertainty

Probability starts by admitting you don't know what's going to happen. A coin is about to be flipped, a die rolled, an image about to be classified. Before it lands, you list every way it could turn out. That complete list of possible outcomes is the sample space, written Ω (capital omega).

For a single coin, Ω = {H, T}. For one die, Ω = {1, 2, 3, 4, 5, 6}. Each element is one outcome, one complete and mutually exclusive way the world could be after the experiment.

Think of pulling one card from a shuffled deck. Before you look, you list every card it could be: all 52 of them. That whole list is the sample space, the same idea as writing Ω = {1, 2, 3, 4, 5, 6} for a die. "The card is a heart" is then an event, a 13-card subset of that list.

Where this lives in MLWhen an image classifier picks from Ω = {cat, dog, bird, …}, that list of labels is a discrete sample space, and a question like "is the true label a mammal?" is an event, a subset of the classes. Data augmentation is a random experiment of the same kind: each crop, flip, or colour-jitter is one outcome drawn from a space of possible transformations, and the augmented dataset is a sample from it.
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