PRACTICAL GUIDE
Why a Random Wheel Repeats: Three Small Probability Experiments
Work through a repeated result, a run of three and a no-repeat draw without mistaking a short streak for proof of bias.
Suppose a wheel has four equally weighted entries: Cedar, Maple, Pine and Oak. Oak lands twice. That can feel suspicious, but a repeated result is compatible with independent random selection. Before judging the wheel, ask exactly which event you are measuring.
Experiment one: repeat the previous result
After the first spin, its result is fixed. On the next independent spin, one of the four entries matches it, so the chance of a match is 1/4, or 25%. Oak is no less likely because it just appeared. This calculation assumes all four entries remain eligible with equal weights.
Write a prediction before trying several pairs of spins. Count how many pairs match. Ten pairs are far too few to expect the observed fraction to sit neatly at 25%, and even a larger experiment can fluctuate.
Experiment two: three identical results
The first result can be anything. The second must match it, and the third must also match. The probability is 1 × 1/4 × 1/4 = 1/16, or 6.25%.
That differs from predicting three Oaks before any spin. Then all three results are specified, so the probability is 1/4 × 1/4 × 1/4 = 1/64, about 1.56%. “Any repeated label” and “this particular repeated label” are different questions.
In the probability lab, set up three independent wheels with those four equal outcomes. There are 64 equally likely ordered combinations. Four combinations have the same label in all three positions. Run a batch, add the observed counts for those four rows and divide by the total trials. The lab displays individual combinations; do this combined-event calculation separately.
Experiment three: remove the first result
Return to a single wheel and remove each selected entry. Now a second Oak is impossible until Oak is restored. Each draw changes the eligible list. That is selection without replacement, which answers a different practical need, such as making a presentation order.
What a streak can and cannot tell you
A streak alone does not prove a faulty picker. Equally, one tidy-looking batch does not certify that an implementation is correct. Check the input list and weights first, including duplicate labels. Keep the configuration unchanged while collecting results.
Do not wait for a supposedly “overdue” label or keep restarting after an untidy batch. Independent spins do not compensate for the past. For repeat-free turns, change the selection rule explicitly rather than expecting randomness to remember who has already appeared.