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Understanding Binomial Theory: Discrete Variables, Probability Distributions, and Applications

This review covers the fundamentals of binomial theory, focusing on discrete and continuous variables and how to determine the nature of specific examples, such as the number of heads in coin tosses. Various probability distributions are examined, including examples to clarify valid probabilities. The document also discusses the mean and standard deviation in the context of binomial distributions, providing calculations and implications for different scenarios, including the birthday problem. This guide is essential for mastering the concepts related to binomial distributions in statistics.

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Understanding Binomial Theory: Discrete Variables, Probability Distributions, and Applications

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  1. Ch. 6. Binomial Theory-- Questions

  2. Discrete or Continuous • Review– Are the following discrete or continuous variables? How do you know • Number of credits earned • Heights of students in class • Distance traveled to class tonight • Number of students in class

  3. Recall the 2 coin example • Let X= number of heads • New terminology: We call X a “Random Variable” • Note: this variable is discrete • P(head)= ½ for each coin • P(X=0) = 1/4 • P(X=1)= 2/4 • P(X=2) = ¼

  4. Recall 3 coin example • Let X= number of heads • P(head)= ½ for each coin • P(X=0) = 1/8 • P(X=1)= 3/8 • P(X=2) = 3/8 • P(X=3) = 1/8

  5. Probability Distributions • Are these probability distributions? • Ex 1: P(X=0) = .25, P(X=1) = .6, P(X=2) = .15 • Ex 2 : P(X=0) = .2, P(X=1) = .5, P(X=2) = .1 • Ex 3: P(X=0) = .4, P(X=1) = -0.2, P(X=2) = .8 • Ex 4: P(X=0) = .2, P(X=1) = 0, P(X=2) = .8 • Ex 5: P(X=0) = .4, P(X=1) = .9, P(X=2) = -.1 • Ex 6: P(X=0) = .2, P(X=1) = .9, P(X=2) = -.1

  6. Complement of events • If P(snow today)= .2, • What is the P(not snow)? • How are these events related? • Another ex: If P(pass)=.8, P(fail)=?

  7. Binomial Theory

  8. An example of a binomial table

  9. 3 coin example- binomial theory • Let X= number of heads • P(head)= ½ for each coin • P(X=0) = 3C0 * (1/2) 0 (1/2) 3 = 1/8 • P(X=1)= 3C1 * (1/2) 1 (1/2) 2 = 3/8 • P(X=2) = 3C2 * (1/2) 2 (1/2) 1 = 3/8 • P(X=3) = 3C3 * (1/2) 3 (1/2) 0= 1/8

  10. See p=.5 column for coin problemsSee n= 2, 3 for 2, 3 coin problems

  11. Find the probability of observing 3 successes in 5 trials if p = 0.7.If n=5, P(X=3)= 0.309

  12. Example: On a 4 question multiple choice test with A,B,C,D,E, p=0.2, find P(X=3)

  13. Mean and St. Dev. of a Discrete Probability Distribution is the expected value of x = is the standard deviation of x = See book for some general examples. We will just concentrate on a special case: the binomial theory…

  14. Mean and Standard Deviation of a Binomial Distribution

  15. For binomial problems • Mean= • St Dev = • Example: When tossing 6 coins, n = 6, p(head)=.5, q(tail)= .5, • Mean = 6(.5)= 3 heads • St Dev = • = 1.22

  16. Mean and St Dev Example Calculate the standard deviation of a binomial population with n = 100 and p = 0.3. a). 21 b).9 c). 4.5825 d). 4.41 Answer: C

  17. Birthday problem Let E=probability that at least 2 of us have the same birthday. E complement= ?? Recall: P(E)=1-P(E complement)

  18. Answer to Bday problem • If n=5, • P(E complement)= ___ • So P(E)= ___

  19. For larger n

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