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Binomial Distributions

Binomial Distributions. Calculating the Probability of Success. Contents. How to identify binomial distributions. How to calculate binomial probabilities. When to use Normal approximations for binomial distributions. 1. How to identify binomial distributions. Identification.

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Binomial Distributions

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  1. Binomial Distributions Calculating the Probability of Success

  2. Contents • How to identify binomial distributions. • How to calculate binomial probabilities. • When to use Normal approximations for binomial distributions.

  3. 1. How to identify binomial distributions Identification

  4. Binomial Distribution • Discrete random variable • Define X • S={0, 1, 2, …} • Binomial setting • XB(n, p) • Key idea: Count success!

  5. The Binomial Setting • “Success” or “Failure.” • Probability of success same for each trial. • Trials independent. • Fixed number of trials.

  6. Characteristics • XB(n, p) • Expected Value: • Variance:

  7. 2. How to Calculate Binomial Probabilities Calculations

  8. Probability Calculations Where: k is the desired count, n is the fixed number of trials, p is the probability of success, and (1-p) is the probability of failure.

  9. Example What is the probability of tossing a fair coin five times and getting exactly three heads?

  10. Check for Binomial Setting • Success is flipping a head;failure is flipping a tail. • The probability of flipping heads on a fair coin is 50% each time. • Each flip is independent. • There is a fixed number of trials.

  11. Define Values In our example: k = 3 n = 5 p = 0.5 & (1-p) = 0.5

  12. Calculations

  13. More Calculations

  14. Interpretation There is about a 31% chance of flipping a fair coin 5 times and getting exactly 3 heads.

  15. Binomial Distribution Using similar calculations,we can find each probability:

  16. 3. When to use Normal approximations. Normal Approximations

  17. Normal Approximations If n is large enough, XB(n, p)  XN(,). Follow two “rules of thumb:” • np  10, & • N(1-p)  10

  18. The End

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