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This page sets out the method for probability in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Probability topic page.
When you see a bag, spinner, dice or pack of cards with equally likely outcomes, count the favourable ones and the total.
When you see a table of probabilities with a gap, or 'find the probability it does not happen', use the total of 1.
0 is impossible. 1 is certain. 1/2 is even chance.
Every probability sits between 0 and 1.
If a fair spinner has 5 equal sections, each has probability 1/5.
Count the successful sections over the total sections.
3 out of 5 is the fraction 3/5.
You can write it as a decimal or a percentage later. The fraction is enough here.
The probability something does not happen is 1 minus the probability it does.
If red is 1/5, not red is 4/5.
“Likely” is above 1/2. “Unlikely” is below 1/2.
Those words are not exact values unless the question gives numbers.
The probabilities of all the outcomes add to 1.
If they do not, an outcome was missed or double-counted.
Draw a line with 0 on the left and 1 on the right. Impossible events sit on 0. Certain events sit on 1. An even chance sits on 1/2, halfway along the line. A probability of 1/10 is close to 0, so the event is unlikely. A probability of 9/10 is close to 1, so the event is likely. The line stops at 1. There is no place on it for a number such as 3/2.
A probability can also be a decimal or a percentage, but the fraction is the exact value. 1/2 is 0.5, which is 50%. 1/4 is 0.25, which is 25%. In these questions, give the fraction in lowest terms. Before you finish, check the result is not below 0 and not above 1. If it is, the count of outcomes has gone wrong.
Method: 1. Mark 0 as impossible and 1 as certain. 2. Place 1/2 in the middle for an even chance. 3. See whether your fraction is below or above 1/2. 4. Reject any value below 0 or above 1.
Counting works when each outcome has the same chance. A fair spinner with 8 equal sections gives each section a probability of 1/8. If 3 of those sections are shaded, the probability of shaded is 3/8. You do not add the 3 to the 8. The 3 is the numerator and the 8 is the denominator. The fraction is already in lowest terms, because 3 and 8 share no common factor bigger than 1.
If a common factor is hiding, divide the top and the bottom by it. Suppose 6 sections out of 15 are blue. The probability of blue is 6/15. Both 6 and 15 divide by 3, so 6/15 = 2/5. The value has not changed. 2/5 is just the simplest name for the same chance. Simplify after you have built the fraction, not before you have counted.
Method: 1. Count the favourable outcomes. 2. Count every equally likely outcome. 3. Write the favourable count over the total. 4. Cancel the highest common factor.
Questions sometimes use words as well as numbers. Impossible means 0. Certain means 1. Even chance means 1/2. Unlikely means greater than 0 but less than 1/2. Likely means greater than 1/2 but less than 1. The words are bands, not extra single values. Both 1/5 and 1/10 are unlikely, but 1/10 is the smaller chance of the two.
Match each word to a fraction before you choose. Landing on a colour that is not on the spinner is impossible, so the probability is 0. A fair coin landing heads is an even chance, so the probability is 1/2. Rolling a number less than 7 on an ordinary dice is certain, so the probability is 1, because every face is from 1 to 6. Use the word that matches the fraction, and keep the fraction if the question asks for a number.
Method: 1. Work out the fraction before you pick a word. 2. Compare the fraction with 0, 1/2 and 1. 3. Choose impossible, unlikely, even chance, likely or certain. 4. Give the fraction as well if a number is required.
Every probability sits between 0 and 1. An impossible event has probability 0 and a certain event has probability 1. An even chance is 1/2, which is also 0.5 and 50%. Events with a probability close to 0 are unlikely and events close to 1 are likely. Exam questions ask you to mark a cross on a scale from 0 to 1, or to choose from a list of fractions, decimals and percentages. Change them to the same form first, so that 3/4, 0.75 and 75% are seen as the same point.
To place an event on the scale, work out its probability as a fraction first. For a fair spinner with 8 equal sections of which 2 are red, the probability of red is 2/8 = 1/4, which is a quarter of the way along the scale. Never leave a final answer as "2 out of 8" alone. Write 1/4, 0.25 or 25%, and never write a value bigger than 1 or smaller than 0.
Method: 1. Count the outcomes that match the event, and count all the equally likely outcomes. 2. Write the probability as a fraction, favourable over total. 3. Simplify, or change to a decimal or percentage if the question asks. 4. Check the answer is between 0 and 1.
Tip: A probability can be written as a fraction, a decimal or a percentage, but never as a ratio with a colon.
All the possible outcomes of one event have probabilities that add up to 1. So if a counter is red, white or blue, then P(red) + P(white) + P(blue) = 1. When one probability is missing from a table, add the ones you know and subtract the total from 1. If red is 0.25 and blue is 0.45, the sum is 0.7 and the missing probability is 1 - 0.7 = 0.3.
A common twist says that two outcomes are equally likely, such as "the number of orange sweets is the same as the number of purple sweets". Subtract the known probability from 1, then split what is left into two equal halves. If green is 0.4, the remaining 0.6 is shared, so each of the other two colours has probability 0.3. To find a number of sweets, multiply the probability by the total number: 0.3 × 50 = 15. Check that all the probabilities add up to 1.
Method: 1. Add the probabilities you are given. 2. Subtract that total from 1 to find what is left. 3. If two outcomes are equally likely, halve what is left. 4. To find a number of items, multiply the probability by the total number of items.
Some questions do not give the number you need. In a raffle with 150 tickets, 48 were sold to children and 57 to women, and the rest were sold to men. First find the missing frequency: 150 - 48 - 57 = 45. Then the probability that the winning ticket was bought by a man is 45/150, which simplifies to 3/10. Write the subtraction as a line of working, because the method mark is for it.
Always check what the bottom of the fraction should be. For a random pick from all the tickets, the denominator is the total number of tickets. If the question picks from a smaller group, such as "a woman is picked at random", the denominator is the size of that group. Cancel at the end using the highest common factor of the top and bottom, and give the fraction in its simplest form unless the question says otherwise.
Method: 1. Find any missing number by subtracting from the total. 2. Decide whether the pick is from everyone or from a smaller group, and use that group as the denominator. 3. Write the probability as a fraction. 4. Simplify using the highest common factor.
Some questions ask whether one event is more likely than another. You must work out the probabilities and compare them in the same form. If Anna wins a game with probability 3/8, then she loses with probability 1 - 3/8 = 5/8. Since 5/8 is bigger than 3/8, she is more likely to lose. A bare statement without the numbers earns no marks, and the question often says "You must work out the probability".
Explanation questions test the idea of equally likely. Taylor says "The probability of blue is 1/2 because there are two colours." That is only true if red and blue are equally likely. With 5 red and 7 blue counters out of 12, the probability of blue is 7/12, which is not 1/2. Write the reason as a sentence using the numbers, such as "there are not equal numbers of each colour".
Method: 1. Work out each probability as a fraction with the same denominator. 2. Compare the fractions, not just the words. 3. Write a conclusion that names the more likely event. 4. For an explanation, mention that equally likely outcomes need equal numbers or equal chances.
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