Formula sheet
Probability of an event
P(A) = n(A) / n(S) is given on the SPM formula sheet. You must know that it assumes every outcome is equally likely, and that the real work is listing the sample space to get n(S) and counting the favourable outcomes n(A) correctly.
What the symbols mean
- P(A) the probability of event A, a number from 0 to 1
- n(A) the number of outcomes favourable to event A
- n(S) the total number of equally likely outcomes in the sample space S
Given in the exam, or memorise?
P(A) = n(A) / n(S) is given on the SPM formula sheet. You must know that it assumes every outcome is equally likely, and that the real work is listing the sample space to get n(S) and counting the favourable outcomes n(A) correctly.
Why it works
When all outcomes are equally likely, the chance of an event is simply the fraction of outcomes that count as success.
- List the sample space S, all the equally likely outcomes, and count them to get n(S).
- Identify which of those outcomes make event A happen, and count them to get n(A).
- The probability is the ratio n(A) / n(S), the proportion of the sample space in A.
- This value always lies between 0 (impossible) and 1 (certain).
Worked example 1
A bag holds 12 marbles: 5 red, 4 blue and 3 green. One marble is drawn at random.
Find the probability that it is green.
- Sample space: n(S) = 5 + 4 + 3 = 12 equally likely marbles.
- Favourable outcomes (green): n(A) = 3.
- Apply the formula: P(A) = n(A) / n(S) = 3 / 12.
- Simplify the fraction: 3/12 = 1/4.
Worked example 2
Cards numbered 1 to 20 are placed in a box and one is drawn at random. Find the probability that the number is a multiple of 3.
- Sample space: cards 1 to 20, so n(S) = 20 equally likely outcomes.
- Multiples of 3 from 1 to 20: {3, 6, 9, 12, 15, 18}, so n(A) = 6.
- Apply the formula: P(A) = n(A) / n(S) = 6 / 20.
- Simplify the fraction: 6/20 = 3/10.
Where students go wrong
- Using the formula when the outcomes are not equally likely (a biased die or weighted spinner), then this simple ratio does not apply.
- Miscounting n(A), especially with 'and/or', inclusive ranges, or forgetting an outcome such as 0 or a boundary value.
- Giving a probability greater than 1, or forgetting to simplify or convert the fraction as the question asks.
Use it with
- Find the probability of a single event
- Use the addition rule for probability
- Use a tree diagram for combined events
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
Can a probability be greater than 1 or negative?
No. Since n(A) can be at most n(S) and never below 0, P(A) always lies between 0 and 1 inclusive.
A probability of 0 means the event is impossible and 1 means it is certain. If your answer falls outside this range, recheck your counts of n(A) and n(S).
Should I leave the answer as a fraction or a decimal?
Either is accepted unless the question specifies. A simplified fraction like 1/4 is exact and often preferred, while a decimal like 0.25 is fine too.
Avoid leaving an unsimplified fraction such as 3/12, and if you use a decimal, do not round a recurring value like 1/3 too harshly.
What is the sample space and why does it matter?
The sample space S is the complete list of all possible equally likely outcomes of the experiment, and n(S) is how many there are. It is the denominator of every probability, so counting it correctly is essential.
List it clearly first, for a die S = {1,2,3,4,5,6}, giving n(S) = 6.