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C8.1 Introduction to probability

Syllabus
0580–2028–2029
Topic
C8.1
Level
Core

Learning objectives

C8.1.1Understand and use the probability scale from 0 to 1.• Understand and use the probability scale from 0 to 1.C8.1.2Calculate the probability of a single event.• Calculate the probability of a single event.C8.1.3Understand that the probability of an event not occurring = 1 – the• Understand that the probability of an event not occurring = 1 – the probability of the event occurring. Probability notation is not required. Probabilities should be given as a fraction, decimal or percentage. Problems may require using information from tables, graphs or Venn diagrams (limited to two sets). e.g. The probability that a counter is blue is 0.8. What is the probability that it is not blue?

Read and use the probability scale

Probability measures how likely an event is on a scale from 0 to 1 inclusive. A larger number means the event is more likely; every valid probability lies between the two endpoints.

Probability Meaning
00 impossible
between 00 and 0.50.5 unlikely
0.50.5 equally likely to occur or not occur
between 0.50.5 and 11 likely
11 certain

For a fair six-sided spinner numbered 1 to 6, landing on 4 has probability 16\tfrac16, so its arrow belongs one sixth of the way from 0 to 1. Landing on an even number has probability 36=0.5\tfrac36=0.5, while landing on 9 has probability 0 because 9 is not an outcome.

Convert a fraction to a decimal when the scale is marked in decimal intervals. For example, 35=0.6\tfrac35=0.6, so place it just to the right of the midpoint. Compare probabilities by their values, not by the size of the event's label.

A probability cannot be negative or greater than 1. The scale describes likelihood, not the number of times an event must occur in a short experiment; observed frequency is treated in C8.2.

Calculate the probability of one event

When all possible outcomes are equally likely, the probability of one event is the number of favourable outcomes divided by the total number of possible outcomes.

\text{probability}=\frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}

List or count the whole outcome set once, count only outcomes that satisfy the event, form the fraction, then simplify or convert to a decimal or percentage if requested. Information may come from a list, frequency table, graph or a Venn diagram with at most two sets.

A fair seven-sided spinner is numbered 1 to 7. The prime outcomes are 2,3,5,72,3,5,7, so there are 4 favourable outcomes out of 7 and the probability is 47\tfrac47. For red and green balls in the ratio 2:32:3, the probability of red is 22+3=25\tfrac{2}{2+3}=\tfrac25.

The favourable-over-total rule needs equally likely individual outcomes. Do not divide by only the favourable count, and do not assume that categories are equally likely when a table or spinner gives unequal weights. Your final fraction, decimal or percentage must represent the same value.

Find the probability that an event does not occur

An event and its 'not' event are complements: exactly one of them must occur, so their probabilities add to 1.

\text{probability of not }A=1-\text{probability of }A

Given Complement calculation Result
fraction 710\tfrac7{10} 1−7101-\tfrac7{10} 310\tfrac3{10}
decimal 0.470.47 1−0.471-0.47 0.530.53
percentage 18%18\% 100%−18%100\%-18\% 82%82\%

With counts, subtract the event count from the total before dividing. In a table, graph or two-set Venn diagram, first identify every region belonging to the event; everything else is its complement. This avoids missing outcomes outside a named set.

If the probability that a train is late is 0.150.15, the probability that it is not late is 1−0.15=0.851-0.15=0.85. The check is 0.15+0.85=10.15+0.85=1.

Not AA means every allowed outcome outside AA, not an unrelated event you choose. Subtract from 1, not from 100 unless you are working in percentages, and keep the result between 0 and 1.