Yes—a probability can be exactly 0 or exactly 1. In an ordinary finite school probability model, 0 represents an impossible event and 1 represents a certain event. The probability scale includes both endpoints: 0 ≤ P(event) ≤ 1.
The useful repair is to stop treating the scale as if it contains only “possible chances” between the endpoints. First define the sample space, then ask whether the event contains no possible outcomes, some outcomes or all outcomes.
Try this now: roll an ordinary six-sided die. Ask for the probability of rolling a 7 and of rolling a number from 1 to 6. The first event has probability 0; the second has probability 1. Ask your child to explain which outcomes are included.
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1. Place events on the inclusive probability scale
OpenStax’s basic probability guide states that probability lies between 0 and 1 inclusive. “Inclusive” matters: the endpoints are valid probabilities, not values just outside the usable scale.
| Probability | School interpretation | Die example |
|---|---|---|
| 0 | Impossible in the stated model | Roll a 7 |
| Between 0 and 1 | Possible but not certain | Roll a 4 |
| 1 | Certain in the stated model | Roll a number from 1 to 6 |
A probability below 0 or above 1 signals an invalid model or calculation. In percentage form, the same boundaries are 0% and 100%. In fraction form, they are 0 and 1, not “no answer” and “more than certain”.
The event must be read within the model. If the die is not ordinary, or the outcome labels differ, rebuild the sample space instead of importing the familiar six-outcome list.
2. Build 0 and 1 from a sample space
For equally likely finite outcomes, probability is favourable outcomes divided by total possible outcomes. An impossible event has zero favourable outcomes, so its probability is 0 divided by the positive total, which equals 0. A certain event includes every outcome, so numerator and denominator match and the probability is 1.
Original illustration: a bag is stated to contain three red counters and two blue counters. Drawing a green counter has probability 0 in that model. Drawing a red or blue counter has probability 5/5 = 1.
Do not infer certainty from what happened once. Drawing red in one trial does not make “red” certain. Probability 1 belongs to an event that contains all allowed outcomes under the model, not merely the outcome just observed.
Likewise, an event can be very unlikely without having probability 0. A one-in-a-thousand event remains possible. Ask whether the event has any allowed outcome, not whether it feels surprising.
3. Separate impossible, unlikely and certain
Everyday language often uses impossible loosely. Probability work needs a defined experiment and sample space. “A school will close tomorrow” may be judged extremely unlikely from background information, but a simple die-style probability cannot be invented without a model.
In finite school models, probability 0 marks no possible outcomes and probability 1 marks all possible outcomes. In advanced continuous probability, a single exact value can have probability 0 without being logically impossible; that subtlety should be introduced only when the learner’s course reaches continuous distributions.
Original contrast: on a fair spinner divided into four labelled sectors A–D, landing on E is impossible in the stated model. Landing on A is possible. Landing on one of A–D is certain. The words come from set membership before arithmetic.
This prevents the child from using 0 as “I think it will not happen” and 1 as “I am confident”. Mathematical probability is tied to a model and evidence, not mood.
4. Diagnose calculation and language errors
- Excludes 0 and 1: underline the word inclusive in 0 ≤ P(E) ≤ 1.
- Calls a rare event impossible: find whether any favourable outcome remains.
- Calls an observed event certain: separate one result from the full sample space.
- Produces 7/6: check whether favourable outcomes were double-counted or the denominator is wrong.
- Changes the experiment mid-answer: restate the sample space before calculating.
A quick diagnostic uses three cards labelled A, B and C. Ask for events with probability 0, 1/3 and 1. The learner should name the outcomes as well as the numbers.
If the child gets the fraction but cannot explain the event, keep the sample-space diagram. If the explanation is correct but arithmetic fails, the repair is different.
5. Match the boundary idea to the learner’s level
For a Primary learner, use small finite sample spaces and simple fractions. For secondary Mathematics, link the boundary check to complements, combined events and probability models only when those topics belong to the actual course.
Use eduKate’s deeper guides on probability as a model of uncertainty and how probability works when the learner is ready to move beyond endpoint classification.
Bring one sample-space error and the unrevised working to tuition. The diagnosis-first approach in eduKate’s published 3-pax Maths teaching reference identifies whether the gap lies in reading the event, listing outcomes or calculating.
Use the learner’s actual Mathematics subject and level, not Posting Group as a worksheet label. MOE’s Full SBB infographic separates school-entry Posting Groups from subject levels. For examination preparation, verify subject, year and candidate type; SEAB states that SEC begins in 2027.
A calm next step
Bring one impossible event, one possible event and one certain event from the same sample space. Ask eduKate about suitable Maths learning support and confirm level fit and current arrangements. The next step should be testable, without any grade promise.
Sources and scope
Probability and policy references were checked on 11 October 2026. The probability-0 discussion is intentionally scoped to ordinary finite school models; continuous probability requires a more advanced interpretation. All examples and routines are original teaching illustrations or practical guidance, not official examination questions or outcome claims. No grade result is promised.