notes day 6: bernoulli trials
TRANSCRIPT
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:
