The idea, step by step
Ask what changed
Events are dependent when knowing an earlier result changes the chance of a later result. If a drawn ball is kept out, the contents of the bag change. The next draw is from a different collection. State which ball was removed before deciding how the chance changed.
Return and mix again
If the ball is returned and the bag is mixed, the original contents are restored. The next colour chances are the same as before. Separate tosses of a fair coin are also independent when tossed normally: earlier heads do not make a tail due next. We use these words to describe the relationship, not whether events occur close together.
A common trap
A tail is due because a fair coin has shown several heads.
Instead: A normal independent toss starts again with the same two chances. Previous heads do not force a tail.
See the thinking
No replacement
A bag has three red and two blue balls. One red ball is drawn and kept out. What remains?
- Remove the known red ball.
- Two red and two blue balls remain.
- The colour chances have changed to equal.
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?
- The returned ball restores the original collection.
- The bag again has three red and two blue.
Check your understanding
Does keeping a drawn counter out of a bag change the next collection?
Show answer
Yes.
There is one fewer counter, so the earlier draw can affect later chances.
Your turn
6 questions · 9 marks · Work at your own pace.
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 ↗