Overview
This lecture introduces the binomial distribution, explains how it models the number of successes in repeated independent trials, and demonstrates its use with several practical examples.
The Binomial Distribution: Key Concepts
- The binomial distribution models the number of successes in n independent Bernoulli trials.
- A Bernoulli trial has two possible outcomes: success (probability p) and failure (probability 1-p).
- Trials must be independent, and the probability of success p must remain the same for each trial.
- The random variable X counts the number of successes in n trials and takes integer values from 0 to n.
Binomial Probability Mass Function (PMF)
- The probability of exactly x successes in n trials is given by:
P(X=x) = nCx × p^x × (1-p)^(n-x), where nCx is the binomial coefficient (combination formula).
- The PMF accounts for the probability of one arrangement of successes and multiplies by the number of such arrangements.
Mean and Variance
- The mean (expected value) of X is np.
- The variance of X is np(1-p).
- The standard deviation is the square root of the variance.
Examples and Applications
- Flipping a coin 100 times and finding the probability of at least 60 heads uses the binomial distribution.
- Buying a lottery ticket weekly for 4 weeks and winning exactly twice is also binomial.
- Rolling a die three times, with success defined as rolling a 5, has n=3, p=1/6, and is binomial.
- For n=3, p=1/6, the probability of exactly 2 fives is calculated as 3C2 × (1/6)^2 × (5/6)^1 ≈ 0.0694.
- Counting the number of surviving 90-year-old Canadian males out of 20, with p=0.82, is binomial.
- The probability that exactly 18 out of 20 survive is 20C18 × 0.82^18 × 0.18^2 ≈ 0.173.
- To find the probability at least 18 survive, sum the probabilities for 18, 19, and 20, giving a total of ≈ 0.275.
Key Terms & Definitions
- Binomial Distribution — Probability distribution of the number of successes in a fixed number of independent Bernoulli trials.
- Bernoulli Trial — An experiment with exactly two outcomes: success or failure.
- Success Probability (p) — Probability that a single trial results in success.
- Binomial Coefficient (nCx) — Number of ways to choose x successes out of n trials.
- Mean (μ) of Binomial — Expected number of successes, calculated as np.
- Variance (σ²) of Binomial — Variability in the number of successes, calculated as np(1-p).
Action Items / Next Steps
- Practice calculating binomial probabilities for different n and p values.
- Review the mean and variance derivations for binomial distributions.
- Prepare to use the binomial distribution in future probability and statistics problems.