How to Calculate Probability
Build a bag of marbles, draw without looking, and watch the chances change. A free interactive tool for primary and early secondary school maths.
Try it: build your bag and draw
Set the marble counts, decide whether to put each one back, and see how the chances shift on every draw. Then use the predictor below to work out 'and' / 'or' questions step by step.
Marble bag
Put marbles in the bag, then pull one out without looking. The chance of each colour depends on how many of that colour are in the bag — and whether you put the marble back before drawing again.
12 marbles in the bag
Chance of each colour on the next draw
Bag contents
12 / 30 marblesDraw history will appear here.
Predict the chance
Drag a marble onto a circle (or tap to add it). Each column is one draw from the bag. Put two marbles in the same column to say "either colour works".
or = +·and = ×
Marbles to drag — 12 marbles left in the bag
Drop a marble into a slot to start.
Without replacement — chances change
Each marble stays out once it's drawn, so the bag changes. The chance of each colour on the next draw goes up or down depending on what came out.
What is probability?
Probability is just a way of putting a number on chance. Some things definitely happen — a coin that's flipped will come down. Some things definitely don't — a marble that isn't in the bag can't be drawn out. Probability gives every event in between a number that says how likely it is.
Probability is a fraction between 0 and 1
A probability is always a number between 0 and 1. 0 means the event can't happen at all. 1 means it has to happen. Anything in between is a fraction.
Try the bag above. Add five red marbles and seven blue marbles, and the chance of drawing red is 5 out of 12 — written as the fraction 5/12. The chance of drawing blue is 7/12. Add the two together and you get 12/12, which is exactly 1 — because one of those two colours has to come out.
Example: Bag with 5 red and 7 blue marbles → P(red) = 5/12.
The probability scale: impossible to certain
You'll see five words used a lot when probability is described in plain English:
- Impossible — probability of 0. A bag with no green marbles will never give you green.
- Unlikely — probability less than 1/2.
- Even chance — probability of exactly 1/2. A fair coin landing heads.
- Likely — probability more than 1/2.
- Certain — probability of 1. A bag with only red marbles will always give you red.
These five words turn up on every primary-school assessment. Once you know the fraction, the word that goes with it is just a label.
Counting the marbles in the bag
The whole game is counting. How many marbles match what you want? That's the top of the fraction (the numerator). How many marbles total? That's the bottom (the denominator). Divide one by the other and you have the probability. Every marble has the same chance of being picked, so all you have to do is count.
How to calculate probability
Once you can count the marbles, every probability question follows one of three patterns: a single event, two events with "and" between them, or two events with "or" between them.
Simple events: favourable ÷ total
For a single draw, you're done as soon as you've counted. Count the favourable outcomes (marbles of the colour you want), divide by the total (all marbles in the bag), and that's the probability.
Example: Bag with 5 red, 4 blue, 3 green → P(red) = 5/12, P(blue) = 4/12 = 1/3, P(green) = 3/12 = 1/4.
Two events and the AND rule (multiply)
When a question asks for the chance that one thing and another thing both happen, you multiply. The chance of two heads in a row from a fair coin is 1/2 × 1/2 = 1/4. The chance of drawing red then red from the bag — with replacement, so the bag stays the same — is P(red) × P(red).
Example: Bag with 5 red, 7 blue, draw two with replacement → P(red, red) = 5/12 × 5/12 = 25/144.
Notice the answer is smaller than either fraction you started with. Multiplying two fractions always makes a smaller one — and that fits: needing two separate things to both happen is harder than needing just one, so the chance goes down.
Two events and the OR rule (add)
When a question asks for the chance that one thing or another thing happens — and the two things can't happen at the same time — you add. Drawing one marble gives you exactly one colour, so red and blue are mutually exclusive. Their probabilities add up.
Example: Bag with 5 red, 4 blue, 3 green → P(red or blue) = 5/12 + 4/12 = 9/12 = 3/4.
Notice the answer is bigger than either fraction on its own. Adding two fractions makes a bigger one — and that fits: if any one of the outcomes counts, you have more ways to succeed, so the chance goes up.
A quick way to remember: "and" is ×, "or" is +.
With replacement vs without replacement
This is where probability questions split into two camps — and where most marks are lost. The wording is small but the maths is different.
With replacement: the bag stays the same (independent events)
If you put the marble back before the next draw, the bag is exactly the same as before. Every draw has the same probabilities. The two draws are independent — what happened on the first one doesn't change what's possible on the second.
For a bag with 5 red and 7 blue, the chance of drawing red twice in a row, with replacement, is 5/12 × 5/12 = 25/144. The 5/12 is the same both times because nothing about the bag changed.
Without replacement: the bag changes (dependent events)
If you keep the marble out, the bag is smaller and the colour counts have shifted. The second probability is calculated from the new bag. The two draws are dependent — the first outcome affects what's possible on the second.
Bag with 8 red and 2 blue. Draw red first → bag now has 7 red and 2 blue. The chance of red again on the second draw is 7/9, not 8/10.
Example: Bag with 8 red and 2 blue, draw red then red without replacement → 8/10 × 7/9 = 56/90 = 28/45.
Why this matters on scholarship tests and Year 9 maths
ACER, Edutest and AAS scholarship maths sections love coloured-counter problems. Year 9 of the Australian curriculum formally introduces with/without replacement. Sometimes the question says it ("with replacement", "without replacement"); sometimes you infer — "draws two marbles" usually means without replacement (you can't draw a marble that's already in your hand). Read the question twice before you answer.
Worked example: drawing a red, then a blue (without replacement)
Bag has 8 red marbles and 2 blue marbles. Draw two marbles, one after the other, without putting the first one back. What's the probability of drawing red then blue?
- 1
Set up the problem
Start with 8 red marbles and 2 blue marbles — 10 marbles total. We want the probability of drawing red on the first pull and blue on the second pull, without replacing the first marble in between.
- 2
Find P(red) on the first draw
There are 8 red marbles out of 10 total. P(red, first) = 8/10. This simplifies to 4/5.
- 3
Update the bag
The red marble is now out. The bag has 7 red marbles and 2 blue marbles left — 9 marbles total. The denominator dropped by one. So did the red count.
- 4
Find P(blue) on the second draw
From the updated bag, there are 2 blue marbles out of 9 remaining. P(blue, second after red) = 2/9.
- 5
Multiply for the AND of two draws
To get the chance of both events happening, multiply the two probabilities along the path: P(red then blue) = 4/5 × 2/9 = 8/45, which is about 17.8%. Key takeaway: when one draw changes the bag, calculate the second probability from the changed bag — then multiply.
Common mistakes to avoid
"I keep drawing red, so blue must be 'due' next time"
This is the gambler's fallacy. When you draw with replacement (or flip a coin, or roll a die), every go is independent — the marble went back, so the bag is exactly the same and nothing is ever "due". A run of reds doesn't make blue any more likely. Without replacement is the one case where the chance really does change after a draw — but that's because the bag physically lost a marble, not because luck evens out.
Mixing up "and" with "or"
"And" means multiply, "or" means add. P(red and then blue) multiplies the two chances; P(red or blue on one draw) adds them. Build both in the predictor above and watch which operation the page uses — multiply for a sequence, add for alternatives.
Forgetting whether you put the marble back
This is the single biggest mistake on two-draw questions. Sometimes the question says it outright; sometimes you have to read it from the wording. "Draws two marbles" almost always means without replacement — you can't draw a marble that's already in your hand.
Where probability goes next
The bag is one way into probability. Once you're comfortable with it, the same ideas show up everywhere — dice rolls, coin flips, spinners, weather forecasts, sports stats. From here you can move into tree diagrams (a drawing of every possible outcome), the sample space for two dice (36 outcomes), and the difference between theoretical and experimental probability (what the maths says versus what actually happens after many trials).
For curriculum-specific revision, jump into our Year 5 or Year 6 maths guides, browse the full Australian Curriculum index, or start a free practice test to try the kinds of probability questions that come up in school assessments and scholarship exams.
Probability vocabulary
The words that turn up most often in probability questions — and what each one means in plain English.
- Probability
- A number from 0 to 1 that says how likely something is to happen.
- Outcome
- One possible result of a draw (for example, pulling out a red marble).
- Sample space
- The full list of every possible outcome (red, blue, green if the bag holds those three colours).
- Event
- One or more outcomes you're interested in (for example, "drawing red or blue").
- Draw
- A single pick from the bag.
- Independent events
- One result doesn't change the chance of the next (with-replacement draws).
- Dependent events
- One result changes the chance of the next (without-replacement draws).
- With replacement
- You put the marble back before the next draw, so the bag stays the same.
- Without replacement
- You keep the marble out, so the bag changes for the next draw.
- Mutually exclusive
- Two events that can't both happen at once (a single marble is red or blue, not both).
- Favourable outcomes
- The outcomes you're counting (the red marbles, if you want P(red)).
- Fair (or unbiased)
- Every marble has the same chance of being picked — what "drawing without looking" means.
Frequently asked questions
What is probability?
A number from 0 to 1 that tells you how likely something is. 0 means it can't happen; 1 means it definitely will. Anything in between is a fraction — like 5/12 for drawing red from a bag of 12 marbles where 5 are red.
How do you calculate probability?
Count the outcomes you want (favourable), then divide by the total number of outcomes. P(A) = favourable ÷ total. For a bag of 5 red and 7 blue, P(red) = 5/12.
What's the difference between with and without replacement?
With replacement, you put the marble back, so the bag stays the same and every draw has the same chances. Without replacement, you keep the marble out, so the bag changes and the chances change with it.
How do you calculate the probability of two draws?
Multiply the chance of the first by the chance of the second. With replacement the chances stay the same; without replacement, use the bag's new counts after the first draw.
What does "P(red or blue)" mean?
Add the two chances. If the bag has 5 red, 4 blue and 3 green, then P(red or blue) = 5/12 + 4/12 = 9/12 = 3/4. Add for "or" (mutually exclusive), multiply for "and".
Are the marble draws fair?
Yes — the tool picks one marble at random each draw, with every marble having an equal chance. That's exactly what "drawing without looking" means in probability questions.
Is probability on the ACER, Edutest or AAS scholarship test?
Yes — at the Year 6 and Year 7 entry level, probability and chance questions appear in the maths or numerical reasoning section. With/without replacement on coloured-counter problems is a classic.
What year do you learn probability in the Australian curriculum?
From Foundation up: chance vocabulary first (likely / unlikely), listing outcomes by Year 5, sample spaces by Year 7, and with/without replacement formally introduced in Year 9 — the concept this tool demonstrates.