Introduction to Probability
Define outcomes and events, calculate probabilities with a stated model, and distinguish equally likely outcomes from observed frequencies.
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Probability starts with a model of possible outcomes
Probability expresses uncertainty on a scale from 0 to 1. An outcome is one possible result; the sample space is the collection of outcomes; an event is a set of outcomes satisfying a condition.
For a fair six-sided die, the sample space is 1 through 6. The event “even” is 6. Under the stated equal-probability model, P(even) = 3/6 = 0.5. Counting favorable outcomes only works this way when the outcomes being counted are equally likely.
Worked example: two dice are not one uniform sum
Two independent fair dice have 36 equally likely ordered pairs. The sum 2 arises only from (1,1), while sum 7 arises from six pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
Thus P(sum = 2) = 1/36, and P(sum = 7) = 6/36 = 1/6. The 11 possible sums are not equally likely. The model assumptions—fair dice and independent rolls—justify assigning probability 1/36 to each ordered pair.
Complements simplify “at least one”
For an event A, its complement Aᶜ means A does not occur:
For two independent fair coin tosses, the probability of at least one head is 1 minus the probability of no heads. No heads means TT, with probability 1/4. The answer is 3/4, not 1/2 + 1/2 = 1: that addition double-counts HH.
More generally, the total probability across mutually exclusive outcomes covering the whole sample space must be 1. Events that overlap require an overlap correction when their probabilities are added.
A probability is not a promise about a short run
A model probability of 0.5 does not require five heads in every ten tosses. Under independent trials with a stable probability, long-run relative frequencies tend toward that probability; short runs still fluctuate. Earlier tails do not make a head “due” on the next independent toss.
An observed proportion is a data summary that may estimate a probability for a defined population or process. A site’s conversion rate last week need not be the same this week if traffic, measurement or the product changes.
Probability 0 also deserves care. In a continuous model, an exact point has probability 0 even though observations take individual values. Zero probability is not universally synonymous with logical impossibility.
Practice
A fair die is rolled. What is the probability of a result at least 5? What is the probability of a result below 5?
Show the explanation
At least 5 means 6: 2/6 = 1/3. Below 5 is the complement: 4/6 = 2/3. These events are disjoint and cover all six outcomes, so their probabilities sum to 1.
Continue
Learn the probability rules and conditional probability. OpenStax introduces independent and mutually exclusive events.
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