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Math · Data and probability
Probability: Impossible, Unlikely, Likely or Certain? Spinners, Bags and Fair Games
Describe chance with words and fractions, compare spinners and bags, catch the 'it's due' mistake and test whether a game is fair. Includes 28 original Grade 3–5 questions with explanations.
The idea
Probability is how we describe the chance that something will happen. Some events are impossible: rolling a 7 on a number cube that only goes from 1 to 6. Some are certain: pulling a red cube from a bag of only red cubes. Most events are somewhere in between, and we call them unlikely, equally likely or likely.
To be more exact, count. First find all the outcomes that are equally likely, such as the 6 faces of a number cube or the 8 equal parts of a spinner. Then count the favourable outcomes, the ones you are looking for. Probability = favourable outcomes ÷ total outcomes. On the spinner in the diagram, blue covers 4 of the 8 equal parts, so the probability of blue is 4/8, or 1/2.
A probability line runs from 0 to 1. Impossible events are at 0, certain events are at 1, and an event with a probability of 1/2 sits right in the middle: equally likely to happen or not. Unlikely events are between 0 and 1/2; likely events are between 1/2 and 1.
Probability tells you what to expect over many tries, not exactly what will happen next. If you flip a coin 10 times, you might get 6 heads instead of 5. That is normal. And a coin does not remember: after four heads in a row, heads and tails are still equally likely on the next flip.
Worked examples
Example 1 · Choose a likelihood word
A bag holds 2 green cubes and 8 orange cubes. How likely is it to pull out green?
- Count the total: 2 + 8 = 10 cubes.
- Count the favourable outcomes: 2 green.
- 2 out of 10 is less than half, but more than zero.
Unlikely (probability 2/10 = 1/5), but not impossible.
Example 2 · Write a probability as a fraction
On the 8-part spinner, what is the probability of landing on red or yellow?
- Red covers 2 parts and yellow covers 1 part.
- Favourable outcomes: 2 + 1 = 3. Total equal parts: 8.
- Write favourable over total.
3/8
Example 3 · Compare two bags by fraction, not by count
Bag A has 2 red and 2 blue. Bag B has 3 red and 5 blue. Which bag gives a better chance of red?
- Bag A: 2 red out of 4, which is 1/2.
- Bag B: 3 red out of 8, which is less than 4/8 = 1/2.
- Bag B has more red cubes, but a smaller fraction of them.
Bag A gives the better chance.
Example 4 · Is the game fair?
Roll a number cube. Player 1 scores on an even number. Player 2 scores on an odd number. Is it fair?
- Even: 2, 4, 6. That is 3 outcomes.
- Odd: 1, 3, 5. That is 3 outcomes.
- Each player has 3/6 = 1/2.
Yes. Both players have the same probability.
Example 5 · Predict over many tries
You pull a cube from the bag in Example 1, put it back, and repeat 50 times. About how many greens should you expect?
- The probability of green is 2/10 = 1/5.
- 1/5 of 50: 50 ÷ 5 = 10.
- Real results vary, so expect a number near 10, not exactly 10.
About 10 greens.
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: An unlikely event can still happen. Impossible means there are zero ways for it to happen.
Why this fails: A bag with more red cubes can still give a worse chance if it has many more cubes in total. Compare fractions.
Why this fails: Three colours does not mean 1/3 each. Probability counts equal parts, so a colour covering half the spinner has probability 1/2.
Why this fails: Coins, cubes and spinners do not remember past results. Each try has the same probability as before.
Why this fails: Probability describes what to expect over many tries. Real results usually land near the prediction, not exactly on it.
Foundation
Foundation practice
Choose the likelihood word. Ask: how many ways can it happen, and how many ways in total?
1. You roll a regular number cube with the numbers 1 to 6. How likely is it that you roll a 7?
- likely
- unlikely
- certain
- impossible
2. A bag holds only red cubes. How likely is it that you pull out a red cube?
- certain
- likely
- equally likely
- unlikely
3. A bag holds 9 blue cubes and 1 yellow cube. How likely is it that you pull out a blue cube?
- likely
- unlikely
- certain
- impossible
4. Using the same bag of 9 blue cubes and 1 yellow cube, how likely is it that you pull out the yellow cube?
- impossible
- equally likely
- unlikely
- likely
5. Look at the spinner in the diagram: 8 equal parts, 4 blue, 2 red, 1 green and 1 yellow. Which colour is the spinner most likely to land on?
- red
- green
- yellow
- blue
6. On the same spinner, compare landing on green with landing on yellow.
- They are equally likely.
- Green is more likely.
- Yellow is more likely.
- Neither can happen.
7. On a number cube with 1 to 6, is rolling an even number more likely, less likely, or equally likely as rolling an odd number?
- More likely
- Equally likely
- Less likely
- It depends on the last roll
Practice
Practice practice
Write probability as a fraction: favourable outcomes over total equally likely outcomes.
8. On the 8-part spinner, what is the probability of landing on blue? Write it in simplest form.
9. On the same spinner, what is the probability of landing on red?
10. A bag has 3 red, 5 green and 2 white marbles. What is the probability of pulling out a green marble?
11. Using the same bag, what is the probability of pulling out a marble that is NOT red?
12. You roll a number cube with 1 to 6. What is the probability of rolling a number greater than 4?
13. Using the bag of 3 red, 5 green and 2 white marbles, order the colours from least likely to most likely.
14. The letters of CANADA are written on six equal cards and mixed up. What is the probability of picking an A?
- 1/4
- 1/6
- 3/4
- 3/6 = 1/2
Think
Think practice
These questions target common mistakes. Explain why the tempting answer fails.
15. Pedro flips a fair coin four times and gets heads every time. What is true about his next flip?
- Tails is more likely because it is 'due'.
- Heads is more likely because it is on a streak.
- Heads and tails are still equally likely.
- It cannot land on heads again.
16. Bag A has 2 red and 2 blue cubes. Bag B has 4 red and 4 blue cubes. Which bag gives a better chance of pulling red?
- Neither; the chances are the same.
- Bag A
- Bag B
- You cannot compare bags of different sizes.
17. Bag A has 3 red and 1 blue. Bag B has 5 red and 5 blue. Which bag gives a better chance of pulling red?
- Bag B, because it has more red cubes
- Bag A
- They are the same
- Bag B, because it has more cubes
18. A spinner is half red, one quarter blue and one quarter yellow. Ava says, "There are three colours, so each colour has a 1/3 chance." What is the probability of red?
19. True or false: 'Unlikely' means it will never happen.
- True
- False
20. Game: roll a number cube. Player 1 scores on 1 or 2. Player 2 scores on 3, 4, 5 or 6. Is the game fair?
21. Change the rule from question 20 so the game becomes fair.
Challenge
Challenge practice
Design, predict and judge results. Show your reasoning.
22. Design a bag of 12 cubes so that the probability of red is 1/4 and blue is twice as likely as green. Use only red, blue and green.
23. You spin the 8-part spinner 40 times. About how many times would you expect it to land on blue?
24. Mila spins the 8-part spinner 20 times and gets red 6 times. Expected: about 1/4 of 20 = 5. Does this prove the spinner is unfair?
- Yes, because 6 is not 5.
- No. 6 is close to 5, and random results vary, especially with only 20 spins.
- Yes, red must be bigger than it looks.
- No, because spinners are always fair.
25. You flip two coins. What is the probability of getting one head and one tail?
- 1/3
- 1/4
- 2/4 = 1/2
- 3/4
26. On a probability line from 0 to 1, where does 'rolling a number less than 7 on a 1-to-6 number cube' belong?
- At 0 (impossible)
- At 1/2
- Between 1/2 and 1
- At 1 (certain)
27. A bag of 20 marbles has a 3/5 probability of giving green. How many green marbles are in the bag?
28. With one number cube, which is more likely: rolling 2, 3 or 5, or rolling a number greater than 3?
- They are equally likely.
- Rolling 2, 3 or 5
- Rolling a number greater than 3
- Neither can happen.
Answer key
- 1. impossible
There is no 7 on the cube, so there are zero ways for it to happen. - 2. certain
Every cube is red, so every pull gives red. - 3. likely
9 of the 10 cubes are blue. It will happen most of the time, but it is not certain because the yellow cube could come out. - 4. unlikely
Only 1 of 10 cubes is yellow. Unlikely does not mean impossible: it can still happen. - 5. blue
Blue covers 4 of the 8 equal parts, more than any other colour. - 6. They are equally likely.
Each colour covers 1 of the 8 equal parts. - 7. Equally likely
Even: 2, 4, 6. Odd: 1, 3, 5. Three ways each. - 8. 4/8 = 1/2
4 blue parts out of 8 equal parts. 4/8 simplifies to 1/2. - 9. 2/8 = 1/4
2 red parts out of 8. 2/8 simplifies to 1/4. - 10. 5/10 = 1/2
There are 3 + 5 + 2 = 10 marbles and 5 are green. - 11. 7/10
10 − 3 = 7 marbles are not red: 5 green and 2 white. - 12. 2/6 = 1/3
Only 5 and 6 are greater than 4. Be careful: 4 itself is not greater than 4. - 13. white, red, green
2/10 < 3/10 < 5/10. The colour with the fewest marbles is least likely. - 14. 3/6 = 1/2
C-A-N-A-D-A has three A's among six letters. Count every card, not each different letter once. - 15. Heads and tails are still equally likely.
A coin has no memory. Each flip still has 1 way to land heads and 1 way to land tails. Tails is not 'due'. - 16. Neither; the chances are the same.
Both bags are half red: 2/4 = 4/8 = 1/2. More red cubes does not help if there are more cubes in total too. - 17. Bag A
Bag A: 3/4. Bag B: 5/10 = 1/2. Bag B has more red cubes, but a smaller fraction of its cubes are red. - 18. 1/2
The parts are not equal in size, so you cannot just count colours. Red covers half of the spinner. - 19. False
Unlikely means it can happen but probably will not. 'Impossible' means it will never happen. - 20. No. Player 2 has 4/6 = 2/3 of the outcomes and Player 1 has only 2/6 = 1/3.
A fair game gives each player the same probability of scoring. - 21. Example: Player 1 scores on 1, 2 or 3; Player 2 scores on 4, 5 or 6.
Any split that gives each player 3 of the 6 numbers is fair, such as even versus odd. - 22. 3 red, 6 blue, 3 green
1/4 of 12 is 3 red. The other 9 must split so blue is double green: 6 blue and 3 green. - 23. About 20 times
Blue has probability 1/2, and 1/2 of 40 is 20. The real number will probably be close to 20, but not always exactly 20. - 24. No. 6 is close to 5, and random results vary, especially with only 20 spins.
Small differences from the expected number are normal. Many more spins would give stronger evidence. - 25. 2/4 = 1/2
List all outcomes: HH, HT, TH, TT. Two of the four have one head and one tail. A common mistake is to list only three outcomes and answer 1/3. - 26. At 1 (certain)
Every number on the cube is less than 7, so 6/6 = 1. - 27. 12
3/5 of 20: 20 ÷ 5 = 4, and 4 × 3 = 12. - 28. They are equally likely.
2, 3 and 5 are three outcomes. 4, 5 and 6 are also three outcomes. Both have probability 3/6 = 1/2, even though the number 5 is in both groups.