The Probability Scale

Activities to try

1. 1

   Placing probabilities on the 0-to-1 measure line

   Why it works: A probability is one point on a bounded measure line; the same value can be written as a fraction, decimal or percentage, and nothing can lie below 0 or above 1.

   1. Work with a 0-to-1 line marked with the fixed landmarks 0 (impossible), 1/2 (even chance) and 1 (certain).
   2. Place a probability of 0.7 and a probability of 1/4 at their correct positions, and read off the same point shown three ways as a fraction, a decimal and a percentage.
   3. For each placement, state which landmark the event sits near and justify its position ('0.7 is above even chance, so toward the certain end').
   4. Place a set of everyday-event cards on the scale and justify each position in words.
   5. Test the bounds: try to place a marker beyond the certain end or below the impossible end and state why no probability can go there.
2. 2

   Fair vs biased: when symmetry works and when it doesn't

   Why it works: Equal probabilities require equal sections/outcomes. For a biased device symmetry no longer gives the value: if the bias is stated you use the given value (red = 0.7); where neither symmetry nor a stated value is available (a drawing pin), the probability can only be estimated from repeated data. The scale applies in every case.

   1. Start with a fair 3-section spinner (equal sections) and calculate each colour's probability as favourable/total — here the three colours genuinely sit at equal thirds; similarly a fair six-sided die gives each number a probability of 1/6 by symmetry.
   2. Switch to a biased spinner that is STATED to be designed so red comes up 70% of the time, with the red section much larger. Predict where the three colours sit on the line BEFORE checking.
   3. Use the stated 70% to place red, and run or observe repeated spins to see that the three colours do not come out at equal thirds — red's observed proportion tends to sit near 0.7, above even chance and toward the certain end, while blue and yellow together share roughly the remaining 0.3 (they may be close to each other, since their split is not given); a finite run only approximates each value rather than landing on it exactly.
   4. Place P(red) = 0.7 above even chance, toward the certain end, and work out P(not red) = P(blue or yellow) = 0.3 by subtracting from 1, placing it below even chance (blue and yellow share that 0.3, and the split between them is not given).
   5. State in words why 'three colours' did not guarantee '1/3 each' — only equal sections would.
3. 3

   Predict-observe-explain with a real drawing pin

   Why it works: Stating a prediction and reason before testing exposes the 'two ways to land, so even chance' assumption; because the pin is asymmetric and no value is stated, its probability must be estimated from experiment rather than assumed to be a half, and the recorded data give that estimate.

   1. Before any testing, predict the probability that a dropped drawing pin lands point-up, and write down the number AND the reasoning.
   2. Drop the pin 40–50 times, recording point-up and point-down tallies.
   3. Compare the observed proportion to the prediction and note whether point-up and point-down came out roughly equal or clearly different; whatever the tallies show, explain why the pin's lack of symmetry means a value of 1/2 cannot simply be assumed.
   4. Mark the estimated point-up probability on the 0-to-1 line at the position the data settle toward.
   5. Explain what the stated-beforehand prediction revealed about the equiprobability assumption, and why this probability had to be estimated rather than calculated.