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IB DP Mathematics: Applications and Interpretation | SL & HL | Statistics and Probability | Topic 4.8
📋 Learning Intentions
- State the four conditions required for a binomial distribution to apply, and verify whether a given scenario satisfies them.
- Calculate exact binomial probabilities using the formula \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \) and using the TI-84 Plus CE.
- Find cumulative binomial probabilities \( P(X \leq k) \), \( P(X \geq k) \), and \( P(a \leq X \leq b) \) using the GDC.
- Determine the mean \( \mu = np \) and variance \( \sigma^2 = np(1-p) \) of a binomial distribution and interpret them in context.
- Interpret binomial model outputs in real-world contexts, including identifying the most likely outcome and commenting on the suitability of the model.
- Apply the binomial distribution to solve multi-step probability problems in Paper 2 extended-response questions.
Real-World Introduction
A pharmaceutical company tests a new vaccine. Each participant either develops immunity or does not — there are exactly two outcomes. If the vaccine has a 90% success rate and 20 patients are treated, how likely is it that all 20 develop immunity? How likely is it that at least 18 do? These are not guesses — they are precise calculations using the binomial distribution.
The same mathematical structure appears in quality control (how many defective items in a batch?), genetics (how many offspring inherit a trait?), and sport (how many free throws does a player make in 10 attempts?). Whenever a process is repeated a fixed number of times and each repetition has the same two possible outcomes, the binomial distribution is the correct model.
🎯 Analogy — The Coin Flip Factory Imagine a machine that stamps coins. Each stamp either produces a perfect coin (success) or a flawed one (failure), with the same probability each time, and each stamp is independent of all others. The binomial distribution counts how many perfect coins appear in a fixed batch — it is the mathematics of repeated, identical, independent trials with two outcomes.
Core Concepts & Definitions
Before applying any probability distribution, it is essential to verify that the scenario actually fits the model. The binomial distribution has four specific conditions that must all be satisfied. If any one fails, the binomial model is not appropriate and a different distribution must be used.
📖 Key Definition — The Binomial Distribution A discrete random variable \( X \) follows a binomial distribution , written \( X \sim B(n, p) \), if:
- There are a fixed number of trials, \( n \).
- Each trial has