Probability: count the right outcomes

The sample space comes before the fraction

A probability describes how likely an event is under a specified model. For a finite collection of equally likely outcomes, probability equals the number of favourable outcomes divided by the total number. The equally-likely condition matters: counting colours in a bag is not enough if different colours occur on different numbers of objects. Probabilities range from zero to one. An event with probability one is certain within the model; a high probability is not a promise about one particular trial.

Worked example: one random selection

A bag contains four blue tokens, three orange tokens and five white tokens, identical except for colour. One token is selected uniformly at random. There are 12 token outcomes, so the probability of orange is 3/12 = 1/4. The probability of not orange is 9/12 = 3/4. You can also calculate it as 1 - 1/4. Blue or orange describes disjoint outcomes for this one draw, so its probability is (4 + 3)/12 = 7/12. The sample space counts tokens, not the three colour names.

Replacement changes the second draw

Using the same bag, select two tokens without replacement. The probability that both are blue is (4/12) × (3/11) = 1/11. After drawing one blue token, only three blue tokens and 11 total tokens remain. If the first token is replaced and the bag is mixed before the second draw, the probability becomes (4/12) × (4/12) = 1/9. The difference is caused by the sampling procedure. Always read whether objects are returned, whether draws are ordered and what random-selection assumption is stated.

Complements simplify 'at least one'

Imagine tossing a fair coin twice, with independent tosses. The four equally likely ordered outcomes are HH, HT, TH and TT. At least one head occurs in three outcomes, so its probability is 3/4. Alternatively, subtract the probability of no heads: 1 - P(TT) = 1 - 1/4 = 3/4. 'At least one' includes both exactly one and exactly two. The probability of exactly one head is only 2/4. Write out a small sample space when a phrase could be interpreted in more than one way.

Separate independence from mutual exclusivity

Independent events do not alter one another's probabilities. Mutually exclusive events cannot both occur in the same trial. On one token draw, blue and orange are mutually exclusive; learning that the token is blue rules out orange, so these positive-probability events are not independent. Multiplication for sequential events uses the conditional probability of the later event. It reduces to multiplying unchanged probabilities only when independence is justified.

• Use P(A or B) = P(A) + P(B) - P(both) when events overlap.
• Do not assume every named outcome is equally likely.
• Keep fractions exact until the final requested rounding.

Sources & further reading