what is a sample space in probability
Say "probability" and most people think of formulas first: count the cases and divide. There is a far more intuitive way to see probability, though: as area. Imagine spreading every possible outcome over a single board. That whole board is the sample space, and its area is 1, that is, 100%, because whatever happens must happen somewhere on this board. An event then becomes part of the board, and the probability of that event is the area that region takes up. Cover half the board and the probability is 0.5; cover a quarter of it and it is 0.25. On this one picture, almost every rule of probability, unions and intersections, complements, conditional probability, and independence, comes down to adding, subtracting, and slicing areas. Instead of memorizing formulas, look at the areas. In this lesson you draw that board yourself and handle probability as area.
Start by building the board. The collection of every possible outcome is called the sample space, usually written Ω. Roll two dice and the outcomes run from (1,1) to (6,6), 36 in all. Each gold cell in the figure is one outcome, and since all are equally likely, each has probability 136. Add up all 36 cells and you get 1, because something is bound to come up. Outcomes are not always countable, though. They can be continuous, as when you pick any real number between 0 and 1. In that case the sample space is a whole region, such as a unit square, rather than cells. Press the "Continuous" tab and the cells give way to points spread evenly across the square. Countable or continuous, the key is the same: Ω is the board holding every possible outcome, and its total probability is always 1. That total of 1 is the starting point of every calculation.
Now look at events. An event is a collection of outcomes we care about, such as the two dice adding to 7. On the board an event is just a region, and its probability is the area that region takes up out of the whole board. Since the total area was set to 1, the area of the region is the probability. Drag the gold corner to change the size of event A: when it covers half the board, P(A) is 0.5; when it covers only a quarter, it is 0.25. You can check this with points. Of the many points scattered over the board, those that land inside A turn gold, and their fraction appears below next to the area. The two values always stay close. This is the Monte Carlo method, the way computers actually work out complicated probabilities. Seen as an area, or as the fraction of points landing in a region, an abstract number becomes a visible patch of surface.
If events are regions, combining events means combining regions. Overlap two events A and B and press the buttons one by one. A or B, the union A∪B, is the area of the two regions together. You must not count the overlap twice, so P(A∪B) = P(A) + P(B) − P(A∩B): subtract the overlap once. A and B, the intersection A∩B, is the area where the two regions overlap, the case where both happen at once. The complement Aᶜ is what is left of the board after removing the event, so P(Aᶜ) = 1 − P(A). The chosen region is filled in, and its area appears below as a probability. The "or," "and," and "not" that sound confusing in words are, in the picture, just joining, overlapping, and cutting out regions. Venn diagrams exist for a reason: the logic of probability is really the geometry of area.
Now for the most confusing yet most powerful idea: conditional probability. The probability that A happens given that B has happened is written P(A|B). In a picture it is very easy. Learning that B has happened shrinks our whole world down to B. Everything outside B did not happen, so clear it off the stage. Turn on the toggle and the area outside B darkens as the board shrinks to B. Inside that smaller new world, the event occupies only the part A∩B. So P(A|B) = area(A∩B)area(B): divide the intersection by B, the new whole. The denominator has merely changed from the whole Ω, whose probability is 1, to P(B). Conditional probability means taking B as the new whole and measuring probability afresh. When new information arrives, shrink the sample space and recalculate; that is the seed of Bayesian inference and of all statistical learning.
Finally, independence. Two events are independent when knowing that one happened does not change the probability of the other: P(A|B) = P(A). The information "B happened" adds nothing about A. Here is how to build it as a picture. Split the board with the green vertical line and call the left strip A, so P(A) is that strip's width; split it with the orange horizontal line and call the bottom strip B, so P(B) is that strip's height. Drag the two lines and the region where both happen, A∩B, is the gold rectangle at the bottom left, whose area is width times height, P(A) × P(B). That is the heart of independence: P(A∩B) = P(A) · P(B), just multiply. Because horizontal and vertical do not interfere with each other at all, the area of the intersection is simply the product of the two sides. When one side has no influence on the other, as with two coin tosses or the failures of two unrelated parts, probabilities simply multiply. Independence is the most welcome assumption there is: it finishes a probability calculation with a single multiplication.