Secondary 1 · Mathematics · Probability
Probability and sample spaces
List all possible outcomes of single-step chance experiments, assign probabilities to equally likely outcomes and compare them with experimental results.
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These lessons were written around Australian Curriculum strands and matched to Singapore levels by age (Primary 1 is the first year of primary school). They are not aligned to MOE syllabuses or the PSLE, so check them against the syllabus your child is working towards.
Learning goals
Learners will
- List the sample space for a single-step chance experiment.
- Assign probabilities to outcomes and events when the outcomes are equally likely.
- Compare theoretical probabilities with results from a real experiment.
Success looks like
- I can write the sample space for rolling a die or spinning a spinner.
- I can calculate a probability as number of favourable outcomes ÷ total number of outcomes.
- I can run an experiment, find the relative frequency and explain why it may differ from the theoretical probability.
The big idea
A chance experiment is any action with an uncertain result, like rolling a die. Each possible result is an outcome, and the list of all outcomes is the sample space. For a die the sample space is {1, 2, 3, 4, 5, 6}; for a coin it is {heads, tails}.
An event is a collection of outcomes. The event “rolling an even number” contains the outcomes 2, 4 and 6.
When all outcomes are equally likely, the probability of an event is:
- P(event) = number of favourable outcomes ÷ total number of outcomes.
So P(even) = 36 = 12. Probabilities can be written as fractions, decimals or percentages: 12 = 0.5 = 50%. All probabilities lie between 0 (impossible) and 1 (certain).
The probability that an event does not happen is 1 minus the probability that it does. P(not rolling a 6) = 1 − 16 = 56.
The probability we calculate is called the theoretical probability. When we actually do an experiment, we find the relative frequency (also called experimental probability): the number of times the event happened ÷ the number of trials. If a coin lands on heads 28 times in 50 flips, the relative frequency of heads is 2850 = 0.56.
Experimental results rarely match the theory exactly, because chance varies. But the more trials you do, the closer the relative frequency tends to get to the theoretical probability.
Worked examples
Rolling a die
A fair six-sided die is rolled. Find the probability of rolling a number greater than 4.
- Sample space: {1, 2, 3, 4, 5, 6}, so there are 6 equally likely outcomes.
- Favourable outcomes: 5 and 6, which is 2 outcomes.
- P(greater than 4) = 26 = 13.
Answer: 13
A spinner
A spinner has 8 equal sections: 3 red, 4 blue and 1 green. Find P(blue) and P(not green).
- There are 8 equally likely sections.
- P(blue) = 48 = 12.
- P(green) = 18, so P(not green) = 1 − 18 = 78.
Answer: P(blue) = 12 and P(not green) = 78.
Comparing with an experiment
A coin is flipped 50 times and lands on heads 28 times. Compare the relative frequency with the theoretical probability.
- Relative frequency of heads = 2850 = 0.56.
- Theoretical probability of heads = 12 = 0.5.
- 0.56 is close to 0.5. The small difference is due to chance; with more flips we would expect the result to get closer to 0.5.
Answer: The relative frequency is 0.56, close to the theoretical 0.5.
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
What is the sample space when a coin is flipped?
{heads, tails} — A coin has two possible outcomes: heads and tails.
What is the probability of rolling a 5 on a fair six-sided die?
16 — There is 1 favourable outcome (5) out of 6 equally likely outcomes.
When a fair die is rolled, which event is most likely?
Rolling an even number — P(6) = 16, P(even) = 36, P(less than 3) = 26 and P(7) = 0. Rolling an even number is most likely.
A bag holds 3 red, 5 blue and 2 yellow marbles. One marble is taken without looking. What is P(blue)?
12 — There are 10 marbles and 5 are blue: 510 = 12.
A spinner has 10 equal sections numbered 1 to 10. What is the probability of spinning a number greater than 7? Give your answer as a fraction.
3/10 — The numbers greater than 7 are 8, 9 and 10 — that is 3 outcomes out of 10.
What is the probability of NOT rolling a 2 on a fair six-sided die? Give your answer as a fraction.
5/6 — P(2) = 16, so P(not 2) = 1 − 16 = 56. (The outcomes 1, 3, 4, 5 and 6 are favourable.)
A coin is flipped 40 times and lands on heads 18 times. What is the relative frequency of heads? Give a fraction in simplest form.
9/20 — Relative frequency = 1840 = 920 = 0.45.
A bag has 4 green and 6 white counters. A counter is drawn and put back, 50 times. About how many times would you expect a green counter?
20 — P(green) = 410 = 0.4, and 0.4 × 50 = 20.
Watch out for
- Believing that after several tails in a row, heads is “due”. Each flip is separate: the coin has no memory, so P(heads) is still 12.
- Assuming outcomes are equally likely when they are not, such as thinking a spinner with one large red section and one small blue section gives P(red) = 12. Probabilities only come from counting outcomes when they are equally likely.
- Expecting an experiment to match the theory exactly, such as exactly 25 heads in 50 flips. Discuss how results vary and get closer with more trials.
Tips for parents
- Run real experiments together. Combine your results to get more trials and see the relative frequency settle down.
- Ask for probabilities in all three forms — fraction, decimal and percentage — to strengthen number skills.
- Use everyday language: ask the learner where events such as “rain tomorrow” or “rolling a 7 on a die” sit on a 0-to-1 scale.
Go further
- Make an unfair spinner from cardboard with unequal sections. Predict the probabilities from the angles, then test by spinning 100 times.
- Drop a drawing pin or a bottle cap 50 times and record how it lands. Here theory can't help — the only way to estimate the probability is by experiment.
- Use a spreadsheet's random number function to simulate 1000 die rolls and compare the relative frequencies with 16.
About 45 minutes
You'll need
- A six-sided die
- A coin
- A bag with coloured counters or marbles
- Paper and pencil for tallying
Key words
- Outcome
- One possible result of a chance experiment, such as rolling a 4.
- Sample space
- The list of all possible outcomes, often written in braces, such as {1, 2, 3, 4, 5, 6} for a die.
- Event
- One or more outcomes grouped together, such as “rolling an even number”.
- Equally likely
- Outcomes that each have the same chance of happening.
- Probability
- A number from 0 (impossible) to 1 (certain) describing how likely an event is.
- Relative frequency
- The number of times an event happened divided by the number of trials in an experiment.
Still tricky?
For parents
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