In this lesson
Five words for one idea
Probability is the math of saying how likely something is. It rests on a few precise words. The curriculum names five core vocabulary terms for this lesson, and every later idea sits on top of them:
- A set is a collection of objects of any nature. We write it
between curly braces:
{ heads, tails },{ 1, 2, 3, 4, 5, 6 }. - An outcome is any possible result of an experiment. Flipping a coin has two outcomes; rolling a die has six.
- The sample space of an experiment is the set of ALL its
possible outcomes. The sample space of a coin flip is
{ heads, tails }. - An event is a set of outcomes you care about. "Rolling an even
number" is the event
{ 2, 4, 6 }for a six-sided die. - A favourable outcome is any outcome that matches the event. For "rolling an even number," the favourable outcomes are 2, 4, and 6. That's three out of six in the sample space.
- A simple event is an event involving exactly one outcome. "Rolling a 3" is a simple event; "rolling an even number" is not.
Probability quantifies the likelihood that an event occurs.
The probability of an event is a number from 0 to 1 that says how likely it is. The rest of this lesson and the next one are about where that number comes from.
See it: sample spaces of three experiments
Three experiments, three sample spaces
Flip one coin
Sample space
{ heads, tails }
n = 2
Two equally likely outcomes.
Roll one die
Sample space
{ 1, 2, 3, 4, 5, 6 }
n = 6
Six equally likely outcomes.
Draw one card from a four-card deck
Sample space
{ ♥ A, ♦ A, ♣ A, ♠ A }
n = 4
Four equally likely outcomes.
A finite sample space lets you list every possible outcome. Each
listing is a set: the set of all possible results. Counting
the elements of the sample space tells you how many outcomes there
are (n = 2, n = 6, n = 4 for the three panels). Once you know
the sample space, the rest of probability is counting the elements
that match a particular event.
Certain, impossible, equally likely
Some events are guaranteed; others can't happen. The curriculum gives these their own probability values:
- Certain events have probability
P = 1. An event is certain if every outcome in the sample space matches it. "Rolling a number from 1 to 6 on a standard die" is certain, because every face in the sample space satisfies the event. - Impossible events have probability
P = 0. An event is impossible if no outcome in the sample space matches it. "Rolling a 7 on a standard die" is impossible. No face gives 7.
Between the two extremes sit ordinary events with P strictly
between 0 and 1. For a single die roll:
P(rolling a 4) = 1/6(one favourable outcome out of six).P(rolling an even number) = 3/6 = 1/2(three favourable outcomes out of six).P(rolling 5 or less) = 5/6.
When all outcomes in the sample space are equally likely, the probability of an event is the count of favourable outcomes divided by the count of total outcomes. That's the same structure you saw in the previous lesson's sampling work, applied here to probability.
Equal weight isn't automatic
Not every experiment has equally likely outcomes. A six-sided die is designed to be fair, so all six outcomes are equally likely. A spinner with unequal slices is not.
The widget below has four coloured slices. Each slice has an
adjustable weight from 0 to 8; the slice's angular share of the
disc is proportional to its weight. But the spinner won't spin for
free: with every new set of weights, the theoretical column shows
? and the spin buttons stay locked until you type
P(slice 1) yourself — as a fraction or a decimal. Call it,
then make the experiment chase your number.
Spin the spinner
Adjust the four slice weights, then spin. Compare the theoretical probability of each slice (its share of the disc) with the experimental probability after many trials.
Weight 0 on all but one slice makes that slice CERTAIN (P = 1). · Weight 0 on a slice makes it IMPOSSIBLE (P = 0).
Try this sequence — you'll have to call P(slice 1) at every step:
- Keep all four slices at weight 2 (the default). What's your call? Four equal slices split the disc evenly.
- Set slice 1 to weight 8 and the others to 1. Slice 1 now covers most of the disc — the call is its weight over the new total.
- Set every slice to 0 except slice 1. The remaining slice is certain — what number expresses certainty?
- Set slice 1 to 0. Now slice 1 is impossible — and the gate still wants its probability before the others spin.
The pattern: probability scales with the angular share of the disc. Equal slices give equal probabilities; unequal slices give unequal probabilities; the extreme cases (weight 0 or sole weight) reach the limiting values 0 and 1.
Where it shows up in real life
A school carnival ring-toss is the lesson's anchor. The bottles are arranged in a grid; the ring is small enough that landing it over any one bottle is possible but unlikely. If 30 bottles are arranged in a 5 × 6 grid and one is the "prize" bottle, the sample space (where the ring lands) has many possible outcomes, and exactly one is favourable. The theoretical probability depends on how easy it is to land the ring over any bottle versus the prize one, plus a lot of skill that the math doesn't model.
Other examples are everywhere. A deck of cards has 52 outcomes
in its sample space; "drawing a heart" is an event with 13
favourable outcomes (P = 1/4). A weather forecast that says
"60% chance of snow tomorrow" is announcing the experimental
probability based on historical data. That's the next lesson's
topic.
Lottery tickets make the impossibly-rare case concrete:
"matching all six numbers" has a sample space of about 14 million
combinations, with one favourable outcome per ticket. The
probability is 1 / 14 000 000. That's almost 0, but not actually 0,
which is why people still buy tickets.
Worksheet
These aren't graded. Get them right, get them wrong. The goal is fluency with the five core terms and the certain / impossible / equally-likely framework.
Practice · Not graded
MA.7.PRO.1Practice the idea
01 / 09
A fair spinner has three equal slices coloured red, blue, and green. You spin it once. What is the sample space?
Multiple choice: sample space of a three-colour spinner.Show common mistakes
Student says
“'There are two possibilities, so each has probability 1/2.'”
What it reveals
Reduces probability to 'either it happens or it doesn't' and treats those two possibilities as equally likely. Probability requires equally-likely outcomes; 'rolling a 6' vs 'not rolling a 6' are not.
Targeted response
Build the sample space first. For a die roll, the sample space is { 1, 2, 3, 4, 5, 6 }: six equally likely outcomes. 'Rolling a 6' is one favourable outcome out of six, so P = 1/6. Use the SpinnerSampler with one slice at weight 1 and another at weight 5 to make the same point visually.
Student says
“Thinks probabilities can exceed 1 (e.g., 'P(rain) = 150%').”
What it reveals
Confused probability with proportion or count. A probability is always a fraction of the sample space; the maximum is 1 (the event covers EVERY outcome), the minimum is 0 (no outcome matches).
Targeted response
The largest possible event is the whole sample space itself, which has probability 1 (every outcome favours it). There's nothing 'bigger' than the entire sample space. If a calculation gives P > 1, something went wrong: usually counting an outcome twice or using the wrong denominator.
Student says
“Doesn't distinguish 'event' from 'outcome.' Uses the words interchangeably.”
What it reveals
Misses the subset structure of an event. An event is a SET of outcomes (possibly just one, a simple event, or several). The distinction matters because P(event) sums over the favourable outcomes inside it.
Targeted response
An outcome is ONE possibility; an event is a SET of outcomes that share a property. For a die roll, the outcomes are 1, 2, 3, 4, 5, 6. The event 'rolling an even number' = { 2, 4, 6 } is a set of three outcomes. P(event) is computed by counting the outcomes in the event, then dividing by the size of the sample space.
Going further
The next lesson computes probabilities in two ways: from the theoretical formula (favourable / total outcomes, assuming equal likelihood) and from experimental data (favourable / total trials, observed). It asks how the two relate, why the experimental version converges to the theoretical one over many trials, and why a single past trial can't predict the next.
In Grade 8 and 9, you'll learn about independent and dependent events (does one event change another's probability?), probability trees (multi-step experiments like 'flip two coins'), and eventually conditional probability (the probability of an event GIVEN that another already happened). The five core terms in this lesson (set, outcome, event, favourable outcome, simple event) are the building blocks of all of it.