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Binomial Distributions: Overview, Examples, and Calculation Methods

This lecture recap covers probability distributions, expected value, and binomial distributions. Learn how to calculate probabilities using binomial distribution formulas and cumulative probability tables.

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Binomial Distributions: Overview, Examples, and Calculation Methods

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  1. Lecture 5 Dan Piett STAT 211-019 West Virginia University

  2. Test 1 Recap Median Score – 85%

  3. Last Week • Probability Distributions • Expected Value of a Probability Distribution

  4. Overview • Binomial Distributions and Probabilities

  5. Binomial Distribution • Suppose an experiment possesses the following properties: • There are a fixed number of trials, n • Each trial results in one of two possible outcomes (success/failure) • The probability of a success (p) is the same for each trial • The trials are independent of one another • X = Number of Successes • This is a binomial experiment • Note that Binomial Distributions are Discrete (You cannot have 1.9976 successes)

  6. Example: Flipping a Coin 50 Times and Recording the Number of Heads Requirements This Experiment There are a fixed number of trials, n. Each trial results in a success or a failure Same probability of success over each trial The trials are independent of one another X = The number of successes There are n = 50 trials in this experiment Heads = SuccessTails = Failure The probability of getting a heads remains constant Tosses are independent of one another X = number of heads

  7. General Binomial Distribution • Suppose X counts the number of successes in a binomial experiment consisting of n trials. Then X follows a Binomial Distribution • Notation: X~B( n, p ) • B stands for binomial distribution • p stands for the probability of success on a single trial • For the previous example X~B(50,.5)

  8. Formula for a Binomial Distribution

  9. Problem on Board • Assume that the probability of a child developing a particular respiratory illness is as an infant is 15%. A family has two children. Assume that the illness is not contagious. • Does this constitute a binomial experiment? • Find: • The probability that none of the children develop the illness • The probability that exactly 1 child develops the illness • Using the rule that all probabilities must add to 1. Find: • The probability that exactly 2 children develop the illness

  10. Cumulative Binomial Probabilities • The previous formula can be used to find the probability that X equal to exactly some value • What about other probabilities of interest? • X equal to less than some value? • X equal to more than some value? • X is between two values? • How do we do this?

  11. Back to the Previous Example • What is the probability that at most 1 child gets the illness? • At most = less than or equal to • At most 1 child = {0, 1, 2} • P(At most 1 child) = P(X=0)+P(X=1) • Note: The probability of this event is defined as the sums of the probabilities. • Remember that this only works because Binomial Distributions are discrete • This is great, but what if n and x are large?

  12. Same Example, New Problem • Suppose that a small town has 20 infants. What is the probability that 18 or less develop the respiratory disease? • 18 or less = {0, 1, 2, … , 17, 18, 19, 20} • P(18 or less) = P(X=0) + P(X=1) +…+ P(X=18) • We would need to compute 19 probabilities to solve this. • Is there a better way? • Actually, there are two • Using cumulative probability tables • Using our knowledge of Complementary Probabilities

  13. Cumulative Probability Tables • Because of the difficulty of calculating these probabilities (and how common the binomial distribution is). Cumulative probabilities for specific values of n and x have been tabulated. • Note: These tables will be provided on quizzes and exams. • How to read the table: • Find the appropriate n and p value, look for x • This is the probability that X is less than or equal to that value

  14. Example: Less Than Probabilities • We have our town of 20 infants. Find the following probabilities: • At most 5 develop the disease • Less than 8 develop the disease • At most 2 develop the disease • Less than 3 develop the disease

  15. Greater than Probabilities • So we now know how to calculate the probability that X is equal to exactly some value or the probability that X is less than/less than or equal to some value. • What about the probability that X is greater than/greater than or equal to some value? • Think back to complementary probabilities

  16. Headed back to our Example, n=20 • What is the probability that 19 or more children develop the disease? • 19 or more = {0, 1, 2, … , 18, 19, 20} • P(19 or more) = P(19) + P(20) • Remember back to the previous example: P(At most 18) • P(At most 18) and P(19 or more) are complementary events • What does this mean? • P(19 or more) = 1 – P(At most 18) • This can be very effective for probabilities such as: • P(At least 1) = 1 – P(At most 0) = 1 – P(X=0)

  17. Greater than Probabilities • Remember back to our use of the tables for calculating less than or equal to probabilities • We can likewise calculate greater than/greater than or equal to probabilities using the table. • Watch the = • We want to get our greater than probabilities in terms of less than or equal to • For n = 6 • P(X>3) = 1 – P(X<=3) • {1, 2, 3, 4, 5, 6) • P(X>=3) = 1 – P(X<3) = P(X<=2) • {1,2 ,3, 4, 5, 6}

  18. Example: Greater Than Probabilities • We have our town of 20 infants. Find the following probabilities: • 6 or more develop the disease • At least 8 develop the disease • 3 or more develop the disease • At least 4 develop the disease

  19. In-between Probabilities • So far we’ve done • P(X=x), P(X<x), P(X>x) • One more to go (The probability the X is between 2 values) • P(a < =x <= b) • Example with the disease: P(X is between 2 and 6) • Between 2 and 6 = {0, 1, 2, 3, 4, 5, 6, 7, … , 19, 20) • P(X is between 2 and 6) = P(X<=6) – P(X<=1) • Why? • P(X<=6) = P(0) + P(1) + P(2)+P(3)+P(4)+P(5)+P(6) • P(X<=1) = P(0) + P(1) • Subtract these and the 0 and 1 cancel leaving: • P(2)+P(3)+P(4)+P(5)+P(6) • This is what we want

  20. Example: In-between Probabilities • We have our town of 20 infants. Find the following probabilities: • Between 3 and 7 develop the disease • At least 1 develops the disease, but less than 14

  21. Coming back to Exact Probabilities • We can use the cumulative table to find exact probability as well • P(X=2) = P(X<=2) – P(X<=1) • Same logic as the previous examples • P(X<=2) = P(X=0) + P(X=1) + P(X=2) • P(X<=1) = P(X=0) + P(X=1) • Subtract and you are left with P(X=2)

  22. Mean and Standard Deviation of a Binomial Distribution • Expected Value (Mean) of a Binomial Distribution • n*p • Standard Deviation of a Binomial Distribution • Sqrt(n*p*(1-p))

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