Binomial Distribution

Discrete vs. Continuous Distributions

  • Two types of distributions: discrete and continuous.
  • Binomial distribution: a crucial discrete distribution for counting.
    • Example: Assessing the number of defective tires produced in a tire manufacturing plant.

Binomial Distribution

  • One of the most significant discrete distributions in statistics.

Jet Sharing Example

  • A company rents jets to three clients who share the use of a private jet.
  • Each client has the opportunity to use the jet one in five days per week.
  • The company wants to determine the probability that all three clients want the jet on the same day or that more than one client wants the jet.

Jet Sharing Diagram

  • Illustrates decisions made by each client (client one, client two, and client three).
  • Clients make decisions independently and simultaneously.

Probability Calculation: All Three Clients Want the Jet

  • The probability that all three clients request the jet on the same day:
    • Probability for each client = 0.2
    • P("all three") = 0.2 \times 0.2 \times 0.2 = 0.008

Possibilities: Two of Three Clients Want the Jet

  • Possible scenarios:

    • Client one and client two want the jet, client three does not (1, 1, 0).
    • Client one and client three want the jet, client two does not (1, 0, 1).
    • Client one does not want the jet, client two and client three do (0, 1, 1).

  • Probability calculation for each scenario:

    • P(1, 1, 0) = P(1, 0, 1) = P(0, 1, 1) = 0.2 \times 0.2 \times 0.8 = 0.032

  • The probability that two of the three clients want the jet:

    • P("two clients") = 3 \times 0.032 = 0.096

Probability: More Than One Client Wants the Jet

  • Probability that more than one client wants the jet (either two or three clients):
    • P("more than one") = P("all three") + P("two clients") = 0.008 + 0.096 = 0.104
  • If two or three clients want the jet on the same day, the company faces a problem with a probability of 0.104.

Key Quantities of Binomial Distribution

  • n: number of trials (Bernoulli trials).
  • k: number of successes (clients who want the jet).
  • p: probability of success (common for all trials).

Mean and Variance of Binomial Distribution

  • Mean (Expected Value): "E(x) = np
  • Variance: Var(x) = np(1-p)

Connection to Bernoulli Distribution

  • p is the expected value of a Bernoulli distribution.
  • p(1-p) is the variance of a Bernoulli distribution.
  • Binomial distribution is the sum of n Bernoulli distributions.

Summary

  • The binomial distribution is a key concept for statisticians.
  • Next Steps: Calculations in Excel.