4 · IB Maths
Binomial distribution
Count successes in n independent trials, each with probability p. Highlight a value of k to read P(X = k) and P(X ≤ k), and compare the bars with a normal curve.
Predict first
Start from the default settings (n = 10, p = 0.50). Increase the number of trials n to 20, keeping p = 0.50. How does the standard deviation σ change?
σ = √(np(1 − p)). With p = 0.5 this is √(n/4), so doubling n multiplies σ by √2: from 1.58 to 2.24.
Check it in the model: Move the Trials n slider from 10 to 20 and compare the σ readout (1.58 then 2.24).
P(X = k) bars highlighted k normal curve. Click or drag on the chart to choose k.
mean = np, variance = np(1 − p)
normal: N(np, np(1 − p)), P(X ≤ k) ≈ Φ((k + 0.5 − np) ÷ σ)
Try this
- Set n = 10, p = 0.5 and k = 5. Read P(X = 5), then change p to 0.3 and find the new most likely value of X.
- Move n from 10 to 20 with p = 0.5. Check that the mean doubles but σ only grows by a factor of √2.
- Turn on the normal overlay with n = 10, p = 0.1 and then n = 50, p = 0.4. When is the approximation poor, and why?
Exam tip: before using a binomial model, state the conditions: a fixed number of trials, two outcomes, independent trials and a constant probability p of success.
More on 4: Venn diagrams and conditional probability · Discrete random variables · Continuous random variables · Bayes' theorem: medical test · Standard deviation and outliers · Markov chains · The normal distribution · Poisson distribution · Scatter and regression · Tree diagrams and Bayes’ theorem · Box plots and histograms