Probability

Probability assigns each event a number from 0 (impossible) to 1 (certain), with the probabilities of all possible outcomes summing to 1. For equally likely outcomes it is a counting exercise: favourable cases over total cases. The difficulty is never the definition — it is combining events, where two distinct questions get confused with each other.

The first is whether events are mutually exclusive: can they both happen? If not, P(A or B) = P(A) + P(B). The second is whether they are independent: does one happening change the chance of the other? If not, P(A and B) = P(A)·P(B). These are separate properties, and it is worth noticing that mutually exclusive events are strongly dependent — learning that A occurred tells you B did not, which is the opposite of no information. Treating the two words as synonyms is the most common structural error in the subject.

The other persistent error is the gambler's fallacy. A fair coin that has landed heads five times running is still 50/50 on the next toss, because the coin retains no record of what it did. Nothing "evens out" by compensation. What is true is the law of large numbers: over many tosses the proportion of heads approaches 1/2, and it does so by swamping the early imbalance with new data rather than by correcting it. The absolute difference between heads and tails can and typically does keep growing even as the proportion settles.