Combining Two Events
Two rules cover most probability questions:
P(A and B) = P(A) × P(B|A)
P(A or B) = P(A) + P(B) − P(A and B)
For independent events, P(B|A) is simply P(B), so the first reduces to multiplication. The subtraction in the second exists because adding the two probabilities counts the overlap twice.
Independent, Exclusive and Dependent
| Relationship | Meaning | P(A and B) | Example |
|---|---|---|---|
| Independent | Neither affects the other | P(A) × P(B) | Two coin flips |
| Mutually exclusive | They cannot both happen | 0 | A die showing 3 and showing 5 |
| Dependent | One changes the other's chance | P(A) × P(B|A) | Drawing two cards without replacement |
Independent and mutually exclusive are frequently confused. They are close to opposites: mutually exclusive events are strongly dependent, since knowing one occurred tells you the other did not.
The Complement Is Usually Easier
"At least one" problems are almost always simpler in reverse. The chance of at least one six in four rolls is not 4 × 1/6. It is 1 minus the chance of no sixes: 1 − (5/6)⁴ = 51.8%.
This also explains the birthday problem. The chance that two people in 23 share a birthday is 50.7%, because the calculation compares 253 pairs rather than 23 people.
Conditional Probability
P(A|B) reads as the probability of A given that B occurred. It is P(A and B) divided by P(B). The order matters enormously and reversing it is the base rate fallacy: the probability of a positive test given disease is not the probability of disease given a positive test.
With a disease affecting 1 in 1,000 and a test that is 99% accurate, a positive result means roughly a 9% chance of having the disease — because false positives from the 999 healthy people vastly outnumber the true positive from the one sick person.
Odds and Probability
A probability of 0.2 is odds of 1 : 4 against, not 1 : 5. Odds compare successes to failures; probability compares successes to the total. Betting odds of 3 : 1 against imply a probability of 1/4, which is a regular source of confusion.
Frequently Asked Questions
Does a coin become "due" for heads after five tails?
No. Each flip is independent and the chance stays at 50%. Believing otherwise is the gambler's fallacy. The coin has no memory.
What is the difference between "or" in maths and in speech?
In probability, "A or B" includes the case where both occur. Everyday speech often means exactly one. When the exclusive reading is intended, it must be stated.
Can a probability be greater than 1?
No. Any calculation producing a value above 1 contains an error — usually double-counting an overlap that should have been subtracted.