HL · Probability distributions · Groups of 3–4 · Whiteboard · standupandmath.com
Use the simulator. Drop 100 balls. Sketch the histogram on your board. Describe the shape.
Drop 100 balls. Sketch alongside your first. What changed? Where is the peak now?
Try other values of n and p. For each, predict which bin gets most balls before dropping 500. Write your conjecture for the peak in terms of n and p.
Label every branch: R = 0.7, L = 0.3. Write the probability of every complete path as a product (e.g. RRL = 0.7 × 0.7 × 0.3).
Group paths by landing bin. What do you notice about paths to the same bin? Rewrite each path probability using exponents. Count paths per bin. Write one path's probability using only k, n, p.
P(X = k) = (path count) × (one path probability). Verify sum = 1. Extend to n = 1, 2, 3, 4 — write the path counts as a triangle. What pattern connects each row to the one above? How do you write the count for bin k in an n-row board mathematically, using n and k?
Calculate \(P(X = k) = \binom{4}{k}(0.7)^k(0.3)^{4-k}\) for each \(k\). Show all working. Verify probabilities sum to 1.
Set n = 4, p = 0.7, drop 500. Does the histogram match your table?
List the four conditions that must hold. Write the notation \(X \sim B(n, p)\) and explain each symbol.
Set n = 10. Record observed mean for p = 0.3, 0.5, 0.7. Spot the pattern. Conjecture E(X) = ?
Keep n = 10. Vary p from 0.1 to 0.9. Record observed variance. At which p is spread greatest? Why?
For \(X \sim B(4, 0.7)\), compute \(\text{E}(X) = \sum k \cdot P(X=k)\) using your Phase 3 table. Does it match your conjecture?