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Binomial Distribution Problems

Binomial Distribution Problems. What is Binomial Distribution?. Binomial distribution is the discrete probability distribution of the number of successes in a sequence of “n” independent yes/no experiments, each of which yields success with probability “p”

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Binomial Distribution Problems

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  1. Binomial Distribution Problems

  2. What is Binomial Distribution? • Binomial distribution is the discrete probability distribution of the number of successes in a sequence of “n” independent yes/no experiments, each of which yields success with probability “p” An example, the statement in a true/false question is either true or false. The outcomes are mutually exclusive, meaning that the answer to a true/false question cannot be both true and false at the same time.

  3. How would you use it to solve problems? • In order to use binomial probability to solve problems, we first need to construct a binomial probability. Therefore we have to use the Binomial Probability Formula. • The probability distribution of the random variable X is called a binomial distribution, and is given by the formula: • P(X) = Cnxpxqn−x

  4. Formula Breakdown P(X) = Cnxpxqn−x • n = the number of trials • x = 0, 1, 2, ... n • p = the probability of success in a single trial • q = the probability of failure in a single trial • (i.e. q = 1 − p) • Cnx is a combination • P(X) gives the probability of successes in n binomial trials.

  5. EXAMPLE 1 • Hospital records show that of patients suffering from a certain disease, 75% die of it. What is the probability that of 6 randomly selected patients, 4 will recover?

  6. This is a binomial distribution because there are only 2 outcomes (the patient dies, or does not). • Let X = number who recover. • Here, n = 6 and x = 4. Let p = 0.25 (success - i.e. they live), q = 0.75 (failure, i.e. they die). • The probability that 4 will recover:

  7. Histogram of this Distribution • We could calculate all the probabilities involved and we would get:

  8. The histogram (using Excel) is as follows: • out of the 6 patients chosen, the probability that none of them will recover is 0.17798, the probability that one will recover is 0.35596, and the probability that all 6 will recover is extremely small

  9. Questions? References: Lind, D.A., Marchal, W.G., & Wathen, S.A.(2008). Statistical techniques in business and economics (13th ed.). Boston: McGraw-Hill/Irwin. http://www.intmath.com

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