Basic Probability and the Probability Scale

A probability is a number that tells you how likely something is to happen. The something is called an event, such as "the dice lands on 6" or "it rains tomorrow".

Every probability is a number from 0 to 1. You can write it as a fraction, a decimal or a percentage, so \frac{1}{4}, 0.25 and 25\% all say the same thing. No probability is less than 0 or more than 1.

In the exam you are usually asked to mark a probability on a probability scale, or to "Write down the probability" of an event with a dice, a spinner or counters in a bag, for 1 or 2 marks. Harder questions ask you to compare two chances, or to work out how many counters there are from a probability.

• Impossible: the probability is 0.
• Unlikely: between 0 and \frac{1}{2}.
• Evens (an even chance): exactly \frac{1}{2}.
• Likely: between \frac{1}{2} and 1.
• Certain: the probability is 1.

Rolling a 7 on an ordinary dice is impossible, so its probability is 0. Rolling a number less than 7 is certain, so its probability is 1. Getting heads with a fair coin is evens: \frac{1}{2}

Some scales are split into equal parts with no numbers on them. Count the parts between 0 and 1 first. If there are 10 parts, each part is 0.1; if there are 4 parts, each part is \frac{1}{4}.

Each possible result is called an outcome. A dice has 6 outcomes: 1, 2, 3, 4, 5 and 6. When a dice is fair, or a counter is taken at random, every outcome has the same chance. The outcomes are equally likely.

When the outcomes are equally likely:

P(\text{event})=\frac{\text{number of outcomes that fit the event}}{\text{total number of outcomes}}

P(\text{event}) is short for "the probability of the event".

For counters in a bag, count every counter, not the number of colours. A bag with 1 red counter and 9 blue counters has two colours, but P(\text{red})=\frac{1}{10}, not \frac{1}{2}.

For a spinner, count the sectors only if they are all the same size. A sector label can appear more than once, so count every sector with that label. If the sectors are different sizes, they are not equally likely, and counting sectors gives the wrong answer.

Many events are made of several outcomes. List the outcomes that fit, count them, then divide by the total. On a fair dice, "a square number" is 1 and 4, so its probability is \frac{2}{6}=\frac{1}{3}

Read the boundary words carefully. "Greater than 4" does not include 4. "Less than 3" does not include 3. "At least 4" means 4 or more, so it does include 4.

An event with or fits if either part is true. An event with and fits only if both parts are true. When you list the outcomes for "or", count a number that fits both parts only once.

An event either happens or it does not, so the two probabilities add up to 1:

P(\text{not }A)=1-P(A)

where A is the event.

If the probability of rain is 0.3, the probability of no rain is 1-0.3=0.7. If the probability of winning is \frac{2}{9}, the probability of not winning is 1-\frac{2}{9}=\frac{7}{9}

In the same way, the probabilities of all the possible outcomes add up to 1. If a bag holds only red, white and blue beads, then P(\text{red})+P(\text{white})+P(\text{blue})=1

Sometimes the rest is split between two outcomes, with one twice as likely as the other. If the two left add up to 0.45, split 0.45 into 3 equal parts of 0.15. The more likely outcome has 2 parts, 0.3, and the other has 0.15.

To compare two chances, work out each probability and write them in the same form, such as decimals. Do not just compare the numbers of winners: one bag may have more winners but many more counters altogether.

A probability also tells you how many there are of each kind. If a bag holds 50 counters and P(\text{red})=0.3, there are 0.3\times50=15 red counters.

Work backwards in equal parts. If 7 counters are \frac{1}{4} of the bag, the bag holds 4\times7=28 counters. If a probability of 0.1 stands for 7 counters, then 0.3 is 3 lots of 0.1, so it stands for 3\times7=21 counters. Do not just add 2 to the 7.

• Writing a probability as a ratio or in words. Write a fraction, a decimal or a percentage, such as \frac{4}{15}, never 4:15 or "4 out of 15".
• Giving a word when a number is asked for. "Unlikely" is not a probability; give the value, and use the words only when the question asks for a word.
• Treating every colour or number as equally likely. Count every counter or every equal sector, not the number of different colours or labels, and never count sectors that are different sizes.
• Adding instead of multiplying when working back. If 0.1 stands for 7 counters, then 0.3 stands for 3\times7=21 counters, not 9.
• Misreading the probability scale. Count the equal parts between 0 and 1 before you place or read a cross.
• Comparing numbers of winners instead of probabilities. Work out both probabilities in the same form, then compare.