Single-event probability

Probability measures chance, not certainty

Probabilities lie between 0 and 1. Zero represents an impossible event in the model, one a certain event. For a finite sample space of equally likely outcomes, probability is favourable outcomes divided by total outcomes.

A fair six-sided die has sample space 6. The probability of a value greater than 4 is 2/6 = 1/3, since 5 and 6 are the two favourable outcomes. The numbers label outcomes; adding 5 and 6 would not count them.

A sample space must not omit outcomes

For two fair coin tosses, ordered outcomes are HH, HT, TH and TT. They are equally likely. Exactly one head occurs in HT or TH, so its probability is 2/4 = 1/2. The labels “zero heads, one head, two heads” are not three equally likely outcomes; the middle category contains two ordered cases.

A spinner’s labels are not automatically equally likely. If one sector covers half the circle and two cover a quarter each, a uniformly spun angle gives probabilities 1/2, 1/4 and 1/4. Count sectors only when their angular shares are equal.

Complements complete one sample space

If the chance of rain in a stated model is 0.3, the chance of no rain is 1-0.3 = 0.7. This does not imply exactly three rainy days in the next ten. A probability describes uncertainty, not a guaranteed short-run schedule.

Check your understanding 1 Keep the physical outcomes equally likely

A bag contains three red, two blue and one green identical counters. One is drawn uniformly at random. Find the chance of blue and the chance of not green.

Your answer or drawing on paper

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Blue has probability 2/6 = 1/3. Not green has probability 5/6. The three colour names are not equally likely categories; individual counters are equally likely under the draw model.

Check the model

A coin gave seven heads in ten tosses. That observation alone does not prove the next toss has probability 0.7 of heads or that a tail is “due”. A model of fairness and independence needs justification beyond a short run’s count.