notes day 6: bernoulli trials

BERNOULLI TRIALS AND THE BINOMIAL DISTRIBUTIONSo far in our discussion of probability we have learned about

combinations and permutations because they help us find the number of ways a certain event can happen. Using that information we calculate probabilities. Today we are learning a formula that is used for very specific situations. We will start with a definition:

A Bernoulli experiment is a random experiment, the outcome of which can be classified as either a success or failure (e.g., female or male, life or death, non­defective or defective, heads or tails, pass or fail).

A sequence of Bernoulli trials occurs when a Bernoulli experiment is performed several independent times so that the probability of success, p, remains the same from trial to trial.

If the probability of a success = p, and the probability of a failure = q then q = 1­p because the probability of a success and failure must add up to 1.

Binomial Distribution In a sequence of Bernoulli trials we are often interested in the total number of successes and not in the order of their occurrence. If we let the random variable X equal the number of observed successes in n Bernoulli trials, the possible values of X are 0,1,2,…,n. If x success occur, where x=0,1,2,...,n , then n­x failures occur. The number of ways of selecting x positions for the x successes in the x trials is: nCx

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Citation:

These probabilities are called binomial probabilities, and the random variable X is said to have a binomial distribution.

Summarizing,a binomial distribution satisfies the following properties:1. A Bernoulli (success­failure) experiment is performed n times. 2. The trials are independent. 3. The probability of success on each trial is a constant p; the probability of failure is q =1−p . 4. The random variable X counts the number of successes in the n trials.

The formula to find the probability in a binomial distribution is: