Probability
Calculating the likelihood of single and combined events, using fractions and the probability scale.
What is Probability?
Probability measures how likely something is to happen. It is expressed as a number between 0 (impossible) and 1 (certain). In the 11+ exam, you need to calculate probabilities as fractions, understand the probability scale, and sometimes list outcomes.
Probability = number of favourable outcomes / total number of possible outcomes.
Step-by-Step Method
Count the total number of possible outcomes
For a fair dice, there are 6 outcomes. For a bag of coloured balls, count ALL the balls.
Count the favourable outcomes
Count how many outcomes match what you want (e.g. rolling a 3 = 1 outcome, picking a red ball = however many red balls).
Write as a fraction
Probability = favourable / total. Simplify if possible.
Use the probability scale
0 = impossible, 0.5 = even chance, 1 = certain. Probabilities are always between 0 and 1.
For combined events, list all outcomes
Use a sample space diagram or tree diagram to list every possible combination.
Worked Examples
What is the probability of rolling a 3 on a fair dice?
Working
- Total outcomes: 6 (numbers 1 to 6).
- Favourable outcomes: 1 (just the number 3).
- Probability = 1/6.
A bag contains 3 red, 5 blue and 2 green balls. What is the probability of picking a red ball?
Working
- Total balls: 3 + 5 + 2 = 10.
- Red balls: 3.
- Probability = 3/10.
List all outcomes of flipping two coins.
Working
- Coin 1 can be H or T. Coin 2 can be H or T.
- Outcomes: HH, HT, TH, TT.
- There are 4 possible outcomes.
Common Mistakes
Forgetting to count ALL possible outcomes when calculating the total.
Count every single item, not just the type you are interested in.
Not simplifying the probability fraction.
Always simplify: 4/8 = 1/2, 6/10 = 3/5.
Thinking probability can be greater than 1.
Probability is ALWAYS between 0 and 1. If you get more than 1, something is wrong.
Top Tips
- The probability of something NOT happening = 1 – probability of it happening.
- If all outcomes are equally likely, probability = favourable outcomes / total outcomes.
- For two events, the total number of combined outcomes = outcomes of event 1 x outcomes of event 2.
- Probability 0 = impossible, 1/2 = even chance, 1 = certain.
Ready to practise?
Put these techniques into action with our free practice papers.
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