Binomial/geometric distributions - Statistics AP Study Notes

Binomial Distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. The notation is X ~ B(n, p) where n = number of trials and p = probability of success.

Key Requirements for Binomial:

1. Fixed number of trials (n)
2. Each trial has only two outcomes (success/failure)
3. Constant probability (p) for each trial
4. All trials are independent

Binomial Probability Formula: $$P(X = r) = \binom{n}{r}p^r(1-p)^{n-r} = \frac{n!}{r!(n-r)!}p^r q^{n-r}$$

where q = 1 - p (probability of failure)

Mean: μ = np | Variance: σ² = np(1-p) | Standard Deviation: σ = √[np(1-p)]

Geometric Distribution models the number of trials needed to obtain the first success. The notation is X ~ Geo(p).

Geometric Probability Formula: $$P(X = r) = (1-p)^{r-1}p = q^{r-1}p$$

This represents (r-1) failures followed by 1 success on the rth trial.

Mean: μ = 1/p | Variance: σ² = (1-p)/p²

Memory Aid: "Binomial counts Batch of successes; Geometric finds the Goal (first success)"

Cumulative Probabilities: P(X ≤ r) requires summing individual probabilities or using tables. For geometric: P(X > r) = (1-p)^r is particularly useful.

Cambridge Command Words: Calculate requires numerical answers; Determine needs justification; Find the probability that expects clear probability notation.

Binomial in Action: Consider quality control in manufacturing. A factory produces light bulbs with a 5% defect rate. If you randomly select 20 bulbs, the number of defective bulbs follows X ~ B(20, 0.05). This models situations where you're counting successes across multiple attempts.

Think of it like basketball free throws: If a player has 80% accuracy and takes 10 shots, the number of successful shots is binomial. Each shot is independent, probability stays constant, and we're counting total successes.

Geometric in Action: How many job applications must you submit before getting your first interview? If each application has a 15% success rate, the number of applications follows X ~ Geo(0.15). The geometric distribution answers "how long until success?"

Medical Testing Analogy: Screening patients for a rare condition where 2% test positive. Binomial asks: "Out of 50 patients, how many test positive?" Geometric asks: "How many patients must we test before finding the first positive case?"

Key Distinction: Imagine flipping coins:

• Binomial: "Flip 10 times, count the heads" → fixed trials, count successes
• Geometric: "Keep flipping until you get heads" → variable trials, wait for first success

Insurance Applications: Insurance companies use binomial to model multiple claims (e.g., 1000 policies, 3% claim rate). They use geometric to model time until first claim in a new risk category.

Real-World Tip: If the question mentions "first time," "until," or "before," think geometric. If it mentions "out of" or "in a sample of," think binomial.

Both distributions assume independence — crucial in real applications like polling (sampling without replacement from large populations).