Chapter 13 - 1

Chapter 13: Binomial Distributions

Overview

  • Discusses:

    • The binomial setting and distributions.

    • Applications in statistical sampling.

    • Calculation of binomial probabilities.

    • Technology examples related to binomial distributions.

    • Derivation of binomial mean and standard deviation.

    • Normal approximation to binomial distributions.

The Binomial Setting

  1. Fixed number of observations (n): The number of trials is predetermined.

  2. Independence of observations: Outcomes of one observation do not affect others.

  3. Two categories: Each observation results in a "success" or "failure".

  4. Constant probability (p): Probability of success remains the same for all observations.

Example: Tossing a Coin 4 Times

  • Success: Getting heads.

  • Parameters: n = 4, p = ½.

  • Possible outcomes (X) range from 0 to 4 heads.

Sample Space for Coin Tosses

  • Outcomes when tossing a fair coin 4 times:

    • {TTTT, TTTH, TTHT, THTT, HTTT, HHTT, HTTH, HTHT, THTH, TTHH, THHT, HHHT, HHTH, HTHH, THHH, HHHH}

Probability Distribution of Coin Tosses

  • Dividing occurrences by total outcomes (16) gives:

    • P(X = 0) = 1/16

    • P(X = 1) = 4/16

    • P(X = 2) = 6/16

    • P(X = 3) = 4/16

    • P(X = 4) = 1/16

Quiz Example: Guessing on a Test

  • Success: Guessing the correct answer.

  • Parameters: n = 10, p = ¼.

  • **Random guessing probabilities: **

    • P(X = k) for k = 0 to 10.

Binomial Experiment Example

  • Scenario: Choosing 3 eggs from a carton of 6, with 2 cracked.

    • Possible outcomes for cracked eggs (X): 0, 1, 2, 3.

Identifying Binomial Experiments

  • Examples:

    • (a) Rolling a pair of dice 10 times.

    • (b) Sampling 10 flights that were not on-time from an airline.

    • (c) Selecting 4 students from a class to record the number of females.

    • (d) Counting defects in cars from assembly lines.

    • (e) Buying lottery tickets, recording wins.

Sampling from Finite Population

  • Simple random sample from size N population:

    • X ~ Bin(n, p) if N >> n.

    • Ideally: N > 20n for binomial approximation.

Distribution of Counts

  • Samples of size n = 10 from variable population sizes given different numbers of successes.

Learning Outcomes

  • Know:

    • The defining conditions of a binomial random variable.

    • How to determine a binomial setting.

    • The condition for sampling without replacement to be approximately binomial.

Binomial Distribution

  • Definition: Count (X) of successes has binomial distribution B(n, p).

    • Parameters: n (trials), p (probability of success).

    • Outcomes: Whole numbers from 0 to n.

Calculating Binomial Probability

  • For a binomial random variable (X):

    • P(X = k) = ( \binom{n}{k} p^k (1-p)^{n-k} )

    • Binomial coefficient = Number of ways to choose k successes from n trials.

Accounting for Rearrangements

  • Rearranging 2 successes and 5 failures leads to combinatorial calculations.

Calculating with Technology

  • TI-84 Commands:

    • binompdf(n,p,k) - Probability of exactly k successes.

    • binomcdf(n,p,k) - Cumulative probability for 0 to k successes.

Example: Rolling Dice

  • Find probability for rolling two sixes in 5 dice tosses.

Example: On-Time Flights

  • Testing on-time flights: Calculate probabilities for observed and expected outcomes using binomial distributions.

Practice Problems

  • Ela's true/false test guessing strategy: Analyze probabilities of correct answers.

Summary of Learning Outcomes:

  • Key concepts:

    • Binomial settings and probabilities.

    • Calculation methods (manual and digital).

    • Mean and standard deviation for binomial distributions.

The Normal Approximation to Binomial Distributions

  • As n increases, binomial distributions approximate normal distributions.

    • Apply this method under specific conditions (large n, specific p).

Continuity Correction

  • Adjust probabilities for approximation between discrete and continuous variables:

    • Use, e.g., adding or subtracting 0.5 to values.

Final Examples and Practice

  • Situations illustrating normal approximation and continuity correction with realistic scenarios such as flight bookings and college admissions.

Additional Learning Outcomes

  • Application of statistical concepts to real-world scenarios.