Chapter 5: Discrete Probability Distributions

Binomial Distributions

Bernoulli Process

• Definition: An experiment consisting of repeated trials, known as Bernoulli trials.
• Outcomes: Two possible outcomes, labeled as success or failure.
• Properties:
  1. Experiment consists of repeated trials.
  2. Each trial results in a success or failure.
  3. Probability of success (P) remains constant across trials.
  4. Trials must be independent of each other.

Binomial Random Variable

• Definition: The number of successes (X) in N Bernoulli trials.
• Key Point: Success in the problem may not always represent a positive outcome.
  • Example: "Getting sick" might be defined as a success for calculation purposes.

Probability Distribution

• Bernoulli Trial:
  • Success probability = P
  • Failure probability = Q = 1 - P
• Binomial Distribution Equation:
  • f(x) = b(x; n, p) = C(n, x) * (p^x) * (q^(n-x))
  • Use calculator for nCx to simplify computation.

Mean and Variance

• Mean (μ): μ = np
• Variance (σ²): σ² = npq
• Standard Deviation (σ): σ = sqrt(npq)

Sample Scenarios

• Defects vs. non-defects
• Customer activity (buy/not buy)
• Component survival (survive/die)
• Patient recovery (recover/not recover)
• Water impurity (pure/impure)

Example Problem

• Scenario: Probability distribution for number of convertibles sold.
• Given:
  • N = 3 (three cars sold)
  • P = 0.6 (probability of selling a convertible)
  • Q = 0.4 (probability of not selling a convertible)

Steps to Solve

1. Identify Variables:
   • N = 3, P = 0.6, Q = 0.4
   • X values range: 0, 1, 2, 3
2. Probability Distribution Table:
   • Calculate f(x) for each X value using binomial formula.
   • Calculate cumulative probability F(x).
3. Use Binomial Table:
   • Provides F(x) values; verify calculations.
   • Can find f(x) by subtracting F(x-1) from F(x).

Calculation Example

• f(0): Compute using formula gives 0.064
• f(1): Compute or use table, gives 0.288
• f(2): Compute gives 0.432
• f(3): Compute gives 0.216

Cumulative Probabilities

• F(0): 0.064
• F(1): 0.352
• F(2): 0.784
• F(3): 1.000

Conclusion

• Knowing how to calculate both exact and cumulative probabilities is essential.
• Familiarity with binomial tables can simplify the process.
• Practice setting and solving problems both manually and using aids like tables.