Outcomes, events and probability

A fair die has six faces, so the chance of rolling at least a five is 2 out of 6, about 0.333.

The forecast says a 30% chance of rain tomorrow. That number is made up, but you have heard ones like it. It does not tell you whether you will get wet. It tells you something weaker that is still useful: rain is possible, and less likely than not. Will the bus be late? Will the next free throw go in? We cannot say for sure, but we are not completely in the dark either. Probability is the number we use for that in-between state. It runs from 0 to 1. A probability of 0 means the event can never happen, 1 means it always happens, and 0.5 means it is as likely to happen as not.

One die has six outcomes, and "at least a five" is two of them

Statistics books use a small set of words, and they are worth learning exactly. An experiment is something you do that has an uncertain result, such as flipping a coin or rolling a die. Each possible result is an outcome. The list of all possible outcomes is the sample space, written S. For one coin, S = {H, T}. For one die, S = {1, 2, 3, 4, 5, 6}. An event is any group of outcomes you care about, named with a capital letter. "Rolling at least a five" is the event E = {5, 6}.

Rock, paper, scissors makes a good second example. Two players each show one of three hands, so the sample space has 3 × 3 = 9 outcomes, from (rock, rock) to (scissors, scissors). Three of them are draws: both rock, both paper, both scissors. If both players choose at random, the nine outcomes are equally likely and P(draw) = 3/9 = 1/3. Whether real players choose at random is another matter. The counting rule below depends on exactly that assumption.

When every outcome in the sample space is equally likely, probability is a matter of counting: P(A) = (number of outcomes in A) ÷ (number of outcomes in S). For the die, P(E) = 2 ÷ 6 = 1/3, about 0.333. OpenStax uses a dime and a nickel: the sample space is {HH, HT, TH, TT}, four outcomes, and the event "exactly one head" contains HT and TH, so its probability is 2/4 = 0.5. Notice that HT and TH are different outcomes. The dime and the nickel are two separate coins, and keeping track of which coin did what is what makes the four outcomes equally likely.

Flip one coin, then another, and the tree grows a layer

You can write the outcomes as a list, draw a tree diagram, or draw a Venn diagram. Lists are fine for small experiments. Trees help when the experiment happens in stages, such as flipping one coin and then another: each stage adds a layer of branches, and following a path from the root to a tip gives one outcome. Venn diagrams are best for showing how events overlap, which comes up in a later lesson on OR and AND.

The counting rule needs equally likely outcomes, and real objects do not always cooperate. OpenStax reports that students who tested a Belgian one euro coin got heads 56% of the time in 250 trials. A die with scooped-out spots may lean slightly towards some faces, which is why casino dice have their spots filled with paint of the same density as the cube. When outcomes are not equally likely, the probability has to come from long-run experience instead of counting. The next lesson shows what "long run" means.