Mathematics 7–10 · Year 9
Two draws from a bag: with and without replacement, tree diagram against 100 trials
Statistics and probability
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The idea
Whether the first marble goes back changes the probabilities on the second branch of the tree, and the difference shows up in the relative frequencies of a hundred real draws.
What you need
- Opaque bag with 3 red and 2 blue marbles (identical in size) per pair
- Recording sheet for 100 two-draw trials under each rule
How to do it
- Draw the tree diagram for two draws with replacement and compute P(RR), P(RB or BR) and P(BB).
- Draw the tree without replacement, changing the second-branch denominators to 4, and compute the same three probabilities.
- Run 100 trials with replacement (draw, note, return, shake, draw) and 100 without (draw, note, draw, then return both). Tally RR, mixed and BB.
- Compare each tally with 100 times the tree's probability and pool the class.
- Compute the relative frequency of at least one red in each rule and compare with 1 − P(BB).
What you should see
With replacement P(RR) = 9/25 = 0.36, P(mixed) = 12/25 = 0.48, P(BB) = 4/25 = 0.16; without replacement P(RR) = 3/10 = 0.30, P(mixed) = 3/5 = 0.60, P(BB) = 1/10 = 0.10. In 100 trials without replacement the expected count of RR is 30 with a standard deviation of 4.58, so 21 to 39 is ordinary, and with replacement 36 ± 4.8. The pooled class difference between the two rules (about 6 in 100 for RR and 12 in 100 for mixed) exceeds the pair-level noise once 1000 trials are pooled. The learner knows it worked when the pooled relative frequencies fall within 0.04 of the tree values for both rules, which a class following the procedure meets about 97 times in 100 (a 0.03 margin fails about one class in seven).
What changes
- What you change
- replacement rule
- What you measure
- relative frequency of each outcome pair
- What you keep the same
- same bag and marbles
- bag shaken between draws
- no looking
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- Replacement makes no difference to the second draw.
- RB and BR are the same outcome in the tree.
- The probabilities on the second branch are the same as on the first under either rule.
Safety card
Hazards
- marbles on the floor
Controls
- draw over the desk
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 5 Probability A; page read 2026-09-22MA5-PRO-C-01
- Australian Curriculum v9AC9M9P01AC9M9P03AC9M9P02
Sources
The pages the author read to write this activity.