0580

Basic Probability

Probability

Equally likely outcomes

  • With equally likely outcomes, count how many produce the event and divide that count by how many outcomes there are altogether
  • Count the total carefully, since it is the denominator of everything that follows
    • A bag holds 40 counters of which 14 are red, so the probability of drawing a red one is 14 ÷ 40, which simplifies to 7/20
  • Simplify the fraction at the end, since "in its simplest form" is a standing instruction on these papers
  • Information may come from a table, a graph or a Venn diagram rather than from a sentence

Probabilities that must add to 1

  • Every outcome of an experiment together accounts for the whole of the probability, so they add to 1
  • A table with one probability missing is completed by subtracting the others from 1
  • The event not happening takes up whatever is left, so P(not A) = 1 − P(A)
    • Where a train has a 0.28 chance of being late, the chance of it not being late is 1 − 0.28 = 0.72
  • Events that can never occur together are mutually exclusive, and for those the chance of one or the other is simply the sum of the two
  • An event and its complement are always mutually exclusive, which is why they add to 1
Exam tip

"At random" appears in 25 of the 43 papers and means every outcome is equally likely, which is what lets you count. A probability may be a fraction, a decimal or a percentage — the syllabus says so — but never a ratio, and never a value above 1.

Worked example

Completing a probability table

A bag holds beads that are red, blue, green or yellow. A bead is taken at random. The probabilities of red, blue and green are 0.35, 0.2 and 0.15. Find the probability of yellow, and the probability of taking a green or a yellow bead.

Solution:

  • The four probabilities together must come to 1
  • The three given ones total 0.35 + 0.2 + 0.15 = 0.7
  • So the probability of yellow is 1 − 0.7 = 0.3
  • One bead cannot be both green and yellow, so those events are mutually exclusive and their probabilities add
  • P(green or yellow) = 0.15 + 0.3 = 0.45