Independence and repeated trials
A and B are independent when learning that one occurred does not change the probability of the other: P(A ∩ B) = P(A)P(B). Independence is an assumption about the process, not a consequence of events being different. For independent coin tosses, the probability of two heads is 1/2 × 1/2 = 1/4.
Repeated independent trials with the same success probability lead to the binomial model; see Bernoulli, binomial, and Poisson distributions. When information changes the probability, use conditional probability and Bayes’ theorem.
Long-run frequency is not a guarantee
The law of large numbers says sample averages converge toward their expected value under suitable assumptions as the number of trials grows. It does not say short runs must balance out: a streak of tails does not make heads “due.”
How to combine events without confusing the rules
The addition rule is for “A or B”: add their probabilities, then subtract the overlap because it was counted twice. The multiplication rule is for “A and B”: use P(A ∩ B)=P(A)P(B|A). Only for independent events does this become P(A)P(B). For one fair die, let A mean “even” and B mean “greater than 3.” Then P(A)=3/6, P(B)=3/6, and P(A ∩ B)=2/6. So P(A ∪ B)=3/6+3/6−2/6=4/6. The events overlap and are not independent: knowing the result is even changes the chance it exceeds 3. Listing the outcomes first is a useful way to verify the calculation.
These rules rely on a probability model that matches the process. If a die is biased or outcomes are sampled without replacement, equal-likelihood or independence assumptions may fail. Write down what is being randomized before choosing a formula; see conditional probability when what is already known changes the calculation.
Try this above
Change the event probabilities or trial count in the visualiser. Compare a single outcome with the distribution of many repeated trials.
A probability can be specified from a theoretical model, estimated from repeated observations, or assigned from expert information; those are different sources of uncertainty. Long-run relative frequency is one way to assess a model, but a finite sample will not match the probability exactly. For events, complements are often easier to calculate: “at least one success” is 1−P(no successes). This shortcut is especially useful in repeated independent trials. A probability model should describe the sampling mechanism, not just the outcomes observed so far.