Understand the idea.
Ask what changed
Two events are dependent when the earlier result changes the chance of a later result. After a draw without replacement, the chosen ball stays out. Recount the remaining colours before describing the next draw.
Which colour was removed matters. Removing a red ball reduces the red count, while the blue count stays unchanged. The next draw uses the remaining collection, not the original bag.
Return and mix again
Returning the ball and mixing restores the original colour counts before the next random draw. In this model, the next colour chances do not depend on which colour was drawn before; the draws are independent.
Normal separate tosses of a fair coin are independent too. Several heads do not make tails due: the coin has not changed. Independence concerns whether earlier results change later chances, rather than how soon the events occur.
See the method in action.
No replacement
A bag has three red and two blue balls. One red ball is drawn and kept out. What remains?
Worked solution
- 1
Remove the red ball from the red count.
The drawn red ball is kept out, so it is unavailable next time.
- 2
Keep the blue count unchanged.
No blue ball was removed.
- 3
Compare the remaining colours.
Equal colour counts now give equal colour chances for a random draw.
- 4
Identify the relationship between the draws.
The first draw changed the collection and the next colour chances, so the draws are dependent.
Two red and two blue; the draws are dependent.
Replacement
A blue ball is drawn from a bag of three red and two blue balls, then returned before mixing. Have the next colour chances changed?
Worked solution
- 1
Track the blue ball while it is out.
Drawing blue temporarily reduces the blue count by one.
- 2
Return the ball before the next draw.
Replacement restores the blue count; the three red balls were unchanged.
- 3
Mix and compare with the original contents.
The random draw uses the same colour counts as before, so its colour chances have not changed.
No. The colour chances are the same as before.
Try it yourself.
Check your understanding
Does keeping a drawn counter out of a bag change the next collection?
Show answer & reasoning
Yes.
Yes. Keeping the counter out leaves one fewer counter in the bag. The next draw uses this changed collection, so its chances can depend on which counter was removed.
Keep practising
6 worksheet questions · 9 marks
Read the questions online
- A bag has four red and two blue balls. One blue is kept out. What remains?1 mark
- A bag has four red and two blue balls. A drawn ball is replaced and the bag is mixed. What are the contents before the next draw?1 mark
- Are two random colour draws without replacement from a mixed-colour bag generally independent or dependent?1 mark
- A fair coin has shown heads three times. Must the next toss be tails? Explain.2 marks
- One red ball is removed from a bag with three red and two blue balls. Which colour is now more likely?2 marks
- Explain why mixing matters after returning a counter to a bag.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A bag has four red and two blue balls. One blue is kept out. What remains?1 mark
Remove one from the blue count.
Four red and one blue. - A bag has four red and two blue balls. A drawn ball is replaced and the bag is mixed. What are the contents before the next draw?1 mark
Replacement restores the original collection.
Four red and two blue. - Are two random colour draws without replacement from a mixed-colour bag generally independent or dependent?1 mark
The first draw changes the collection for the next.
Dependent. - A fair coin has shown heads three times. Must the next toss be tails? Explain.2 marks
The next ordinary toss remains independent; tails is not owed.
No. - One red ball is removed from a bag with three red and two blue balls. Which colour is now more likely?2 marks
Two red and two blue remain.
Neither; they are equally likely. - Explain why mixing matters after returning a counter to a bag.2 marks
Restoring counts alone does not make a deliberately selected draw fair.
Mixing supports a random draw instead of deliberately choosing a known counter.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum Mathematics 2026
Ordering likelihood; independent/dependent events; repeated trials; comparing variation.
Official curriculum source ↗