1. Mathematical Formula Reference
Equal Slices Probability
P(A) = 1 / nWhere n is the number of equal-sized options.
Weighted Option Probability
P(A_i) = w_i / Σw_jWhere w_i is the item weight over the sum of all weights.
Expected Value
E[X] = Σ (x_i · P(x_i))Sum of outcomes weighted by their respective probability.
Chi-Square Test Statistic
χ² = Σ [ (O_i - E_i)² / E_i ]Compares observed frequencies (O) against expected (E).
2. Case Study: "Is the Wheel Fair?" (Chi-Square Test)
Consider a 4-option decision wheel spun $N = 80$ times. If the wheel is unbiased, the theoretical expected frequency for each option is $E_i = 80 / 4 = 20$ spins.
| Option | Observed (O) | Expected (E) | (O - E)² / E |
|---|---|---|---|
| Option 1 | 22 | 20 | (22-20)² / 20 = 0.20 |
| Option 2 | 17 | 20 | (17-20)² / 20 = 0.45 |
| Option 3 | 24 | 20 | (24-20)² / 20 = 0.80 |
| Option 4 | 17 | 20 | (17-20)² / 20 = 0.45 |
Summing these values yields the test statistic χ² = 0.20 + 0.45 + 0.80 + 0.45 = 1.90. With degrees of freedom df = k - 1 = 3, the critical value at significance level α = 0.05 is 7.815. Since 1.90 < 7.815, we fail to reject the null hypothesis—confirming the wheel is statistically fair!
Teacher's Lesson Plan (45–60 Mins)
- Introduction (10 mins): Define theoretical vs. empirical probability.
- Group Activity (20 mins): Students spin a 4-option wheel 50 times in teams and record frequencies on their tally sheets.
- Statistical Calculation (15 mins): Compute the Chi-Square test statistic for their team's data.
- Class Discussion (10 mins): Compare individual team results against class-wide aggregated data to demonstrate the Law of Large Numbers.