5.2 - The Binomial Distribution

Discrete Probability Distributions

Overview

Discrete probability distributions model scenarios with discrete outcomes, with a focus on the binomial distribution.

Topics

• 5.1 Discrete Random Variables and their Probability Distributions

• 5.2 The Binomial Distribution

• 5.3 Other Discrete Probability Distributions

The Binomial Distribution

• Definition: A discrete probability distribution used for experiments with two outcomes: success and failure.

• Conditions:

  • Fixed number of trials (n).

  • Trials must be independent and identical.

  • Outcomes classified into two events: success and failure.

  • Probabilities of success (p) and failure (q = 1 - p) remain constant.

Binomial Formula

• Random Variable X: Represents the number of successes in n Bernoulli trials.

• Probability of success on any trial is p; probability of failure, q = 1 - p.

Calculating Probability

Example 1:

• Variables: n = 10, p = 0.8, find P(X = 6).

Example 2:

• Experiment: Drawing 5 cards without replacement. Calculate probabilities of face cards.

Example 3:

• Experiment: Coin flips until Tails appears. Calculate probability for Tails on 7th toss.

Properties of the Binomial Probability Distribution

• Expected Value and Variance: Helps analyze expected outcomes and variability in binomial settings.

Coffee Chain Example:

• Promotion analyzing probability of winning prizes with 300 million cups, resulting probabilities will inform the likelihood of winning.

Using a Binomial Table

• A binomial table simplifies the process of calculating probabilities for various outcomes without reapplying the binomial formula.

Expected Value, Variance, and Standard Deviation

• Formulas:

• Expected Value: E(X) = n * p

• Variance: Var(X) = n * p * q

• Standard Deviation: SD = √(n * p * q)

Application Example:

• Calculate expected prizes, variance, and use Chebyshev’s Theorem to predict outcomes based on statistical data.