13.1 Probability of Single Events

Probability is the mathematics of chance, and it is one of the five strands of ICAS Mathematics: Chance & Data. Questions on single events appear at every level, from the Introductory paper (choosing which spinner is most likely to land on red) to Paper D and beyond, where you are expected to give answers as simplified fractions, decimals or percentages. Because personal calculators are not allowed in ICAS, the arithmetic is always gentle — the marks come from setting up the correct fraction, not from heavy computation.

The Language and Scale of Probability

Every event sits somewhere on a scale from impossible to certain:

• Impossible — cannot happen. Probability 0. (Rolling a 7 on an ordinary six-sided die.)
• Unlikely — could happen, but probably will not.
• Even chance — as likely to happen as not. Probability 1/2, 0.5 or 50%.
• Likely — will probably happen, but is not guaranteed.
• Certain — must happen. Probability 1 or 100%. (Rolling a number less than 7 on a die.)

So probability is a number between 0 and 1 inclusive. If you ever calculate a probability greater than 1, or negative, something has gone wrong — this is a useful sanity check on exam day. A probability can be written three ways, and ICAS switches between them freely: as a fraction (3/4), a decimal (0.75) or a percentage (75%).

The Basic Formula

When every outcome is equally likely — a fair die, a fair coin, a well-mixed bag of marbles — the probability of an event is:

P(event) = number of favourable outcomes / total number of outcomes

Worked example 1: a bag of marbles

A bag contains 3 red, 5 blue and 2 green marbles. One marble is drawn at random. What is P(blue)?

• Total marbles = 3 + 5 + 2 = 10
• Favourable (blue) = 5
• P(blue) = 5/10 = 1/2

Always simplify the fraction. ICAS answer options will show 1/2, not 5/10, and part of the skill being tested is simplification.

Worked example 2: a fair die

What is the probability of rolling a number greater than 4 on a standard die?

• Outcomes greater than 4: 5 and 6, so 2 favourable outcomes
• P = 2/6 = 1/3

A common trap is miscounting the favourable outcomes. 'Greater than 4' does not include 4. Read the wording twice: 'at least 4' would include 4, giving 3/6 = 1/2. ICAS uses exactly this kind of wording contrast.

Cards, Spinners and Everyday Setups

ICAS loves standard contexts. Know these counts cold:

For a spinner, check the sectors are equal. If a spinner has 8 equal sectors numbered 1, 1, 2, 3, 3, 3, 4, 5, then P(3) = 3/8 because three sectors show 3 — count sectors, not distinct numbers.

Worked example 3: cards

One card is drawn from a full pack. What is P(ace)? There are 4 aces out of 52 cards: P = 4/52 = 1/13. If the question asks for a decimal, 1/13 is awkward, so ICAS would more likely ask P(red) = 26/52 = 1/2 = 0.5, or give options in fraction form.

Complementary Events

The event 'A does not happen' is called the complement of A. Since something must either happen or not happen:

P(not A) = 1 - P(A)

Worked example 4

The probability that it rains tomorrow is 0.3. What is the probability it does not rain?

P(no rain) = 1 - 0.3 = 0.7 (or 70%).

The complement is also the quickest route when 'not' is easier to count than the event itself. If a bag has 9 blue and 1 white marble, P(not blue) is just 1/10 — one line instead of counting 9/10.

How ICAS Phrases These Questions

• Introductory and Paper A style: 'Which spinner gives the greatest chance of landing on grey?' — compare fractions by comparing favourable sectors.
• Paper D style: 'A letter is chosen at random from the word BANANA. What is the probability it is A?' — count letters: 3 A's out of 6 letters, so 1/2. The trap is answering 1/3 because there are three different letters.
• Reverse questions: 'A bag has only red and blue counters. P(red) = 3/8. If there are 24 counters, how many are blue?' — P(blue) = 5/8, and 5/8 of 24 = 15.

Practise turning each question into a fraction first, then simplify, then convert if the options demand a decimal or percentage.