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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.

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More Discussion of the Binomial Distribution: Comments & Examples

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  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

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