Mathematics 7–10 · Years 7–8
Drawing pins: a chance experiment with no theoretical answer
Statistics and probability
The idea
When the outcomes of a trial are not equally likely, the only way to estimate a probability is from the relative frequency of many trials, and the estimate steadies as trials accumulate.
Safety card
Hazards
- sharp points can pierce skin
- pins on the floor can be trodden on
Controls
- count pins out and back in
- toss into a tray, not onto the desk
- pick up with a magnet or by the head
- no tossing near faces
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
What you need
- 10 identical steel drawing pins (flat head, about 10 mm diameter) per pair, counted out and counted back in
- 1 plastic cup per pair for tossing
- 1 tray or shallow box lid per pair so pins stay contained
- Tally sheet with columns point up and point down
How to do it
- Discuss why a drawing pin is different from a coin: the two ways of landing are not symmetric, so no value can be worked out in advance.
- Each learner writes a personal estimate of the proportion that will land point up.
- Put 10 pins in the cup, shake, and tip them onto the tray. Count point up and point down. That is 10 trials.
- Repeat 10 times so each pair has 100 trials. Record the running relative frequency of point up after each toss of 10.
- Pool the class: with 12 pairs that is 1200 trials. Compare each pair's 100-trial estimate with the pooled estimate.
- Repeat with a different brand or size of pin and compare the pooled estimates.
What you should see
The pooled relative frequency is the class's best estimate of the probability and no theoretical value exists to check it against; what can be checked is the behaviour of the estimates. Whatever the true proportion p, the standard error of an estimate from n trials is sqrt(p(1 − p)/n), so a 100-trial estimate carries a standard error sqrt(1200/100) = 3.46 times that of a 1200-trial estimate. The one pooled figure has no spread of its own to compare with, so the spread that is measured is the one across the 12 pair estimates: their standard deviation, set against sqrt(p(1 − p)/100) computed from the pooled p. The learner knows it worked when those two numbers agree within about 30 per cent (with only 12 estimates the measured spread carries about 20 per cent uncertainty of its own), and when a different pin gives a different pooled value, showing the probability belongs to the object and not to the number of outcomes.
What changes
- What you change
- number of trials (and pin type in the extension)
- What you measure
- relative frequency of point up
- What you keep the same
- same tossing method from the same cup onto the same tray
- pins that land leaning on another pin are re-tossed
- same height of tip
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- Two outcomes means each has probability one half.
- The probability can be worked out by looking at the pin.
- One hundred trials gives the exact answer.
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 4 Probability; the point on observed probability as the relative frequency resulting from repeated trials of a chance experiment is what this entry rests on, while the page defines probability through a finite number of equally likely outcomes and carries no content point on outcomes that are not equally likely, so the unequal pin is the context rather than the content; ACARA AC9M7P01 was dropped because it asks for probabilities to be assigned to the outcomes of a single-stage event, the one thing a drawing pin does not allow, leaving AC9M7P02 to carry the entry; page read 2026-09-23MA4-PRO-C-01
- Australian Curriculum v9AC9M7P02
Sources
The pages the author read to write this activity.