Mathematics 11–12 · Year 12
Where does the spinner stop: a continuous uniform random variable
Random variables: Continuous random variables (Mathematics Advanced, Year 12)
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The idea
The angle at which a fair spinner stops can take any value from 0 to 360 degrees, so probabilities come from areas under a flat density function rather than from counting outcomes.
What you need
- A spinner: a card disc printed with a 0 to 360 degree protractor scale and a pointer on a pin (or a bottle spinning on a marked disc)
- A tally sheet and a spreadsheet
How to do it
- Spin 40 times and record the stopping angle to the nearest degree.
- Draw a histogram with 45 degree bins and a cumulative relative frequency graph.
- Write the density f(x) = 1/360 for 0 <= x < 360 and the distribution function F(x) = x/360.
- Compute P(X < 90), P(100 < X < 160) and the expected value and variance, and compare with the data.
- Explain why P(X = 90) = 0 even though the pointer can stop at 90 degrees.
What you should see
P(X < 90) = 0.25, P(100 < X < 160) = 60/360 = 0.167, E(X) = 180 degrees, Var(X) = 360^2 / 12 = 10 800 and the standard deviation is 103.9 degrees. Of 40 spins about 10 should fall in each quarter, with a binomial standard deviation of 2.7 spins, and the cumulative graph should lie close to the straight line F(x) = x/360. The learner knows it worked when the cumulative relative frequency hugs that straight line and the sample mean is near 180 degrees.
What changes
- What you change
- the spin
- What you measure
- stopping angle
- What you keep the same
- same spinner
- a firm spin of at least one full turn
- angle read from directly above
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- A probability density of 1/360 means each angle has probability 1/360; each exact angle has probability 0 and only intervals carry probability.
- The density can never exceed 1; for a variable spread over [0, 0.5] the uniform density is 2.
Safety card
Hazards
- the pin point
Controls
- cover the pin with a cork or use a pencil point
Note
No chemicals and no 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 Advanced 11–12 Syllabus (2024), Year 12 focus area Random variables; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-07
- Australian Curriculum v9No Australian Curriculum v9 code is listed.
Sources
The pages the author read to write this activity.