The probability scale

Your turn: find the probability

All six faces of a regular six-sided dice are equally likely. Count first, then pick.

A regular six-sided dice is rolled. What is the probability of rolling a multiple of three?

What you need to know

• A probability is a number that measures how likely an outcome is, which is more precise than words like likely and unlikely.
• Probabilities run from 0 to 1: 0 means impossible, 1 means certain, and the size of the number shows how likely.
• Have a goYour friend says: "My chance of finishing my homework tonight is 1.4, so I'm super-certain!" Can 1.4 be a probability?

  No.

  Probabilities only run from 0 to 1, and 1 already means certain, so there is nothing above it to be "super-certain" about.

• One half (50%) is an even chance: exactly midway, and just as likely to happen as not.
• Above one half is more than an even chance (likely); below one half is less than an even chance (unlikely).
• Probabilities can be fractions, decimals or percentages, and 0% to 100% matches 0 to 1.
• Have a goAn event has a probability of 60%. Is it unlikely, an even chance or likely?

  Likely.

  60% is more than 50%, which is one half, so it is more than an even chance.

• With the same denominator, the smaller numerator is the smaller probability.
• With the same numerator, the larger denominator is the smaller probability; otherwise, use a common denominator.
• Have a goWithout converting anything, which is the smaller probability: 3/10 or 3/7?

  3/10.

  The numerators match, so the larger denominator (10) gives the smaller probability. A bigger denominator does not make a bigger probability.

• P(event) = value is shorthand: P(A) = 3/5 means the probability of landing on A is 3/5.
• When all outcomes are equally likely, the probability is the outcomes in the event over the total outcomes.
• Over a suitably large number of trials, a probability is the proportion of times you expect the event, though it may not happen exactly.

The big picture

A probability puts a number on how likely something is, from 0 (impossible) to 1 (certain), with one half as the even-chance line between unlikely and likely. It can be written as a fraction, a decimal or a percentage. You can compare probabilities by matching numerators or denominators or by using a common denominator, find one from equally likely outcomes, and read it as the proportion of times you expect the event over many trials.

⚠ Watch out

Thinking a bigger denominator means a bigger probability, so 3/8 looks bigger than 1/2 because 8 is bigger than 2. Compare properly: 3/8 is less than 4/8, which is one half. And a probability is not a promise of an exact count.

Memory hook

Picture a fence post at the exact middle of the line: one half. Left of the fence is unlikely, right of it is likely, and "more likely" always means "further right".

Check yourself

Cover the page. A probability is 4/9. Explain how you would decide whether that is likely or unlikely, and what it tells you to expect in 900 trials.