Probability and Binomial Distribution

Overview

This lecture reviews probability concepts, the binomial distribution, and the sampling distribution of the sample proportion with practical examples and step-by-step calculations.

Probability Basics

• Probability of an event = (Number of successful outcomes) / (Total number of outcomes).
• Drawing a green marble: probability is 200/500 = 0.4; blue marble: 300/500 = 0.6.
• Multiple draws with replacement means probabilities remain constant for each draw.

Binomial Distribution

• Binomial distribution calculates the probability of a fixed number of successes in n independent trials with probability p.
• Formula: P(X = k) = C(n, k) * p^k * (1 - p)^(n - k), where k = number of successes, n = number of trials, p = probability of success.
• For three draws, probability of at least two green marbles = P(exactly 2) + P(exactly 3) = 0.288 + 0.064 = 0.352.
• For five draws, use the binomial formula for k = 2, 3, 4, 5 and sum the probabilities to get 0.6634.

Sampling Distribution of the Sample Proportion

• For large samples (e.g., n = 100), binomial calculations become inefficient.
• Use the sampling distribution of the sample proportion (p̂) for approximation.
• Central Limit Theorem conditions: np ≥ 10 and n(1-p) ≥ 10 must be satisfied.
• Standardization (z-score) formula: z = (p̂ - p) / sqrt(p(1-p)/n).
• Find area to the right of z-score for "at least" problems by 1 - area to the left.
• For at least 35 green marbles in 100 draws: z = -1.02, probability ≈ 0.8461 or 84.61%.

Key Terms & Definitions

• Probability — likelihood of an event occurring, ranges from 0 to 1.
• Binomial Distribution — probability distribution for the number of successes in n independent trials with constant probability p.
• Sample Space — the set of all possible outcomes of an experiment.
• Central Limit Theorem — for large n, sample proportion distribution approaches a normal distribution if conditions are met.
• Sample Proportion (p̂) — ratio of successes to total trials in a sample.
• Standardization Formula — converts sample proportion to z-score for normal distribution approximation.

Action Items / Next Steps

• Review prior videos for deeper explanations of these topics if needed.
• Practice solving similar probability and binomial distribution problems.
• Review binomial and normal (z-score) calculation steps for exam preparation.