What is probability?
Probability is how likely something (an event) is to happen.
- An outcome is any result that could happen. Rolling a dice has 6 outcomes: 1, 2, 3, 4, 5 and 6.
- You can write a probability as a fraction (½), a decimal (0.5) or a percentage (50%).
- The biggest possible probability is 1, which is 100%. Nothing can be more certain than certain, so “1000% certain” isn’t real maths!
Types of events
| Event | Probability | Example |
|---|---|---|
| Impossible | 0 (0%): it can never happen | rolling a 7 on a normal dice; the sun turning into a black hole tomorrow |
| Unlikely | less than ½ (below 50%), but not impossible | rolling a 6 on a dice |
| Even chance | exactly ½ (50%) | a coin landing on heads; rolling an even number on a dice |
| Likely | more than ½ (above 50%) | picking a red sweet from 9 red and 1 green |
| Certain | 1 (100%): it will definitely happen | Tuesday coming after Monday |
The probability formula
Probability = the outcomes you want ÷ the total number of possible outcomes
- List every possible outcome on paper. Never try to do it in your head.
- Underline the outcomes you want.
- Write it as a fraction: underlined ÷ total, then simplify.
Fair means every outcome is equally likely, like a fair coin or a fair dice.
Dice questions
A normal dice always has 6 outcomes, and each number has a probability of 1/6. Check each number against the question:
| Question | Outcomes (the ones you want are underlined) | Probability |
|---|---|---|
| an even number | 1, 2, 3, 4, 5, 6 | 3/6 = 1/2 |
| a prime number | 1, 2, 3, 4, 5, 6 | 3/6 = 1/2 |
| a factor of 6 | 1, 2, 3, 4, 5, 6 | 4/6 = 2/3 |
| greater than 4 | 1, 2, 3, 4, 5, 6 | 2/6 = 1/3 |
Bags, boxes and cards
- Add up everything to get the bottom number: 3 red + 5 green + 6 blue = 14.
- Count only the ones you want for the top number.
- P(red) = 3/14. P(green or blue) = 11/14.
It works the same for balls, sweets, mugs, cards or shapes: count the total, then the ones that match.
The probability of something NOT happening
P(not X) = (total − the ones you want) ÷ total, or take it away from 1.
P(not rolling a 6) = (6 − 1) ÷ 6 = 5/6. If P(red) = 3/14, then P(not red) = 1 − 3/14 = 11/14.
Simplest form
Divide the top and bottom by the same number until you can’t any more:
- 6/15 → divide both by 3 → 2/5
- 4/8 → divide both by 4 → 1/2
Which gives the best chance? Compare them
To choose, work out each probability and turn it into a percentage.
Which box gives the best chance of a chocolate cupcake?
| Box | Chocolate cupcakes | Probability | About |
|---|---|---|---|
| Box 1 | 4 out of 7 | 4/7 | 57% ← best chance |
| Box 2 | 5 out of 12 | 5/12 | 42% |
| Box 3 | 5 out of 10 | 5/10 | 50% |
Box 3 has the most chocolate cupcakes, but Box 1 gives the best chance.
Fractions and percentages to learn: 1/2 = 50% · 1/3 ≈ 33% · 2/3 ≈ 67% · 1/4 = 25% · 3/4 = 75% · 1/5 = 20% · 1/10 = 10%
Two events: independent, and multiplying
Independent events don’t affect each other. Every roll of a dice or flip of a coin is a fresh start, so work out each one on its own and ignore what happened before.
When a question asks for two things to happen (“this and then that”), multiply the probabilities: tops × tops, bottoms × bottoms.
A dice is rolled twice. What is the probability of a factor of 6, then a number less than 5? P(factor of 6) × P(less than 5) = 4/6 × 4/6 = 16/36 = 4/9
“The same number both times”: the first roll can be anything. Only the second roll has to match it, so the probability is 1/6.
(Events are dependent when the first one changes the second, like taking a sweet from a bag and not putting it back: there’s one fewer sweet for the next pick.)
Watch out! The bottom number is the total of everything, not just “the other colour”. With 3 red and 5 blue, P(red) is 3/8, not 3/5.
Top tip: always write out all the outcomes and underline the ones you want. Check your answer is between 0 and 1, and simplify!
How likely is it? Tap the right box.
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Rolling a 7 on a normal dice
Without looking, Seán takes one counter out of this bag. How likely is it to be yellow?
A dice is rolled. Tap every outcome that is a factor of 6.
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