1 / 19

More Discussion of the Binomial Distribution: Comments & Examples

Learn about the binomial distribution, its requirements, and how to calculate probabilities using real-life examples. Explore the distribution's mean, standard deviation, and common notation. Discover how it applies to the coin flipping problem and other scenarios.

paulamyers
Download Presentation

More Discussion of the Binomial Distribution: Comments & Examples

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. More Discussion of the Binomial Distribution: Comments & Examples l j Thermo & Stat Mech - Spring 2006 Class 16

  2. The Binomial Distribution applies ONLY to cases • where there are only 2possible outcomes: heads or tails, • success or failure, defective or good item, etc. Requirements justifying use of the Binomial Distribution: 1.The experiment must consist ofn identicaltrials. 2. Each trial must result inonly oneof 2 possible outcomes. 3.The outcomes of trials must be statistically independent. 4.All trials must havethe same probabilityfor a particular outcome.

  3. Binomial Distribution The Probability ofnSuccesses out ofN Attempts is: p = Probability of a Success q = Probability of a Failure q = 1 – p, (p + q)N = 1

  4. Mean of the Binomial Distribution l Thermo & Stat Mech - Spring 2006 Class 16

  5. Standard Deviation (s) of the Binomial Distribution 2 1  3

  6. For The Binomial Distribution

  7. Common Notation for the Binomial Distribution • r items of one type & (n – r) of a second type • can be arranged in nCr ways. Here: ≡ • nCris calledthe binomial coefficient • In this notation, the probability distribution is: • Wn(r) = nCrpr(1-p)n-r • ≡ probability of finding r items of one type& • n – r items of the other type. p = probability of a • given item being of one type .

  8. Binomial Distribution: Example • Problem: A sample of n = 11 electric bulbs is drawn • every day from those manufactured at a plant. The • probabilities of getting defective bulbs are random • and independent of previous results. • Probability that a given bulb is defective isp = 0.04. 1. What is the probability of finding exactly three defective bulbs in a sample? (Probability that r = 3?) 2. What is the probability of finding three or more defective bulbs in a sample? (Probability that r ≥ 3?)

  9. Binomial Distribution, n = 11 l Thermo & Stat Mech - Spring 2006 Class 16

  10. Question 1: Probability of finding exactly • three defective bulbs in a sample? • P(r = 3 defective bulbs) = • W11(r = 3) = 0.0076 l Thermo & Stat Mech - Spring 2006 Class 16

  11. Question 1: Probability of finding exactly • three defective bulbs in a sample? • P(r = 3 defective bulbs) = • W11(r = 3) = 0.0076 • Question 2: Probability of finding three or • more defective bulbs in a sample? • P(r ≥ 3defective bulbs) = • 1- W11(r = 0) – W11(r = 1) – W11(r = 2) • =1 – 0.6382 - 0.2925 – 0.0609 = 0.0084 l Thermo & Stat Mech - Spring 2006 Class 16

  12. Binomial Distribution, Same Problem, Larger r Thermo & Stat Mech - Spring 2006 Class 16 l

  13. Binomial Distribution n = 11, p = 0.04 Distribution of Defective Items Distribution of Good Items l Thermo & Stat Mech - Spring 2006 Class 16

  14. The Coin Flipping Problem • Consider a perfect coin. There are only 2 sides, so the probability associated with coin flipping is The Binomial Distribution. • Problem:6 perfect coinsare flipped. What is the probability that they land with n heads&1 – n tails? Of course, this only makes sense if 0 ≤ n ≤ 6! For this case, the Binomial Distributionhas the form: l Thermo & Stat Mech - Spring 2006 Class 16

  15. Binomial Distribution for Flipping 1000 Coins Note: The distribution peaks aroundn = 500 successes (heads), as we would expect ( = 500) l Thermo & Stat Mech - Spring 2006 Class 16

  16. Binomial Distribution for Selected Values of n & p n = 10, p = 0.1 & n =10, p = 0.9 n = 20, p = 0.5 l Thermo & Stat Mech - Spring 2006 Class 16

  17. n = 5, p = 0.1 n = 5, p = 0.5 Binomial Distribution for Selected Values of n & p n = 10, p = 0.5 l Thermo & Stat Mech - Spring 2006 Class 16

  18. n = 5, p = 0.5 Binomial Distribution for Selected Values of n & p n = 20, p = 0.5 n = 100, p = 0.5 l Thermo & Stat Mech - Spring 2006 Class 16

  19. Binomial Distribution for Selected Values of n & p

More Related