Probability of Combined Events
How to Find the probability of a single event
Use this to find the probability of one event as a fraction of the sample space.
Before you start
- Understanding what 'equally likely' outcomes means
- Simplifying a fraction to lowest terms
- Listing outcomes in a clear, systematic way
When to use it
Use this to find the probability of one event as a fraction of the sample space.
The steps
- List or count all equally likely outcomes, this is the sample space.
- Count the outcomes that satisfy the event (favourable outcomes).
- Write probability = favourable ÷ total.
- Simplify the fraction, or convert to a decimal if asked.
- Check the answer is between 0 and 1.
Worked example
Twelve identical cards are numbered 1 to 12. One card is drawn at random.
Find the probability that the number is a multiple of 3.
- List the sample space: the numbers 1 to 12, giving 12 equally likely outcomes.
- Count the favourable outcomes (multiples of 3): 3, 6, 9, 12 → 4 outcomes.
- Write probability = favourable ÷ total = 4 ÷ 12.
- Simplify the fraction: 4/12 = 1/3.
- Check the answer is between 0 and 1: 1/3 is valid.
A second example, with a twist
The sample space is not a single list but all 36 ordered pairs from two dice, so you must count the outcomes carefully. Two fair dice are rolled.
Find the probability that the two numbers add up to 8.
- List the sample space: each die has 6 faces, so two dice give 6 × 6 = 36 equally likely ordered outcomes.
- Count the favourable outcomes where the numbers add to 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes.
- Write probability = favourable ÷ total = 5 ÷ 36.
- The fraction 5/36 will not simplify (5 is prime and does not divide 36), so leave it, or write ≈ 0.14 as a decimal.
- Check the answer is between 0 and 1: 5/36 is valid.
Formulae you may need
Formula pages
Practise this in a KBAT problem
Frequently asked questions
When two dice are rolled, does order matter?
Yes. Treat the two dice as distinct, so (2,6) and (6,2) are separate outcomes.
Counting this way keeps all 36 outcomes equally likely, which is what the probability formula needs. If you merged them, the outcomes would not be equally likely.
Can a probability be bigger than 1?
No. The favourable outcomes are always part of the sample space, so their count can never exceed the total.
That means every probability lies between 0 and 1. If you get an answer above 1, you have miscounted, so recheck your favourable and total outcomes.
Should I give my answer as a fraction or a decimal?
Either is acceptable unless the question specifies. A simplified fraction is usually clearest and exact, while a decimal can help you compare sizes.
If you round to a decimal, keep two or three figures and show the fraction too, so no marks are lost.