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Binomial Probability Distributions
This chapter explores Bernoulli experiments and the probability distributions of binomial random variables.
Learning Objective

Apply Bernoulli distribution in determining success of an experiment
Key Points
 A Bernoulli (successfailure) experiment is performed n times, and the trials are independent.
 The probability of success on each trial is a constant p; the probability of failure is q=1−p.
 The random variable X counts the number of successes in the n trials.
Term

Bernoulli Trial
an experiment whose outcome is random and can be either of two possible outcomes, "success" or "failure"
Example
 At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A "success" could be defined as an individual who withdrew. The random variable is X = the number of students who withdraw from the randomly selected elementary physics class.
Full Text
Many random experiments include counting the number of successes in a series of a fixed number of independently repeated trials, which may result in either success or failure. The distribution of the number of successes is a binomial distribution. It is a discrete probability distribution with two parameters, traditionally indicated by n, the number of trials, and p, the probability of success. Such a success/failure experiment is also called a Bernoulli experiment, or Bernoulli trial; when n = 1, the Bernoulli distribution is a binomial distribution.
Named after Jacob Bernoulli, who studied them extensively in the 1600s, a well known example of such an experiment is the repeated tossing of a coin and counting the number of times "heads" comes up.
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:
(nx)=n!x! (n−x)!
Since the trials are independent and since the probabilities of success and failure on each trial are, respectively, p and q=1−p, the probability of each of these ways is px(1−p)n−x. Thus, the p.d.f. of X, say f(x) , is the sum of the probabilities of these (nx) mutually exclusive eventsthat is,
f(x)=(nx)px(1−p)n−x,x=0,1,2,...,n.
These probabilities are called binomial probabilities, and the random variable X is said to have a binomial distribution .
Wind pollination
These male (a) and female (b) catkins from the goat willow tree (Salix caprea) have structures that are light and feathery to better disperse and catch the windblown pollen.
Probability Mass Function
A graph of binomial probability distributions that vary according to their corresponding values for n and p.
Key Term Reference
 binomial distribution
 Appears in these related concepts: The Hypergeometric Random Variable, Calculating a Normal Approximation, and Categorical Data and the Multinomial Experiment
 distribution
 Appears in these related concepts: Application of Knowledge, Monte Carlo Simulation, and Selling to Consumers
 experiment
 Appears in these related concepts: Experimental Design, Primary Market Research, and Descriptive and Correlational Statistics
 independent
 Appears in these related concepts: Conditional Probability, Unions and Intersections, and Party Identification
 mutually exclusive
 Appears in these related concepts: Disadvantages of the IRR Method, Multiple IRRs, and Solving Systems of Linear Inequalities
 probability
 Appears in these related concepts: Particle in a Box, The Addition Rule, and Rules of Probability for Mendelian Inheritance
 probability distribution
 Appears in these related concepts: Shapes of Sampling Distributions, Probability Distributions for Discrete Random Variables, and Overview of How to Assess StandAlone Risk
 random variable
 Appears in these related concepts: Chance Processes, The Sample Average, and Expected Value
 variable
 Appears in these related concepts: Related Rates, Math Review, and Psychology and the Scientific Method: From Theory to Conclusion
Sources
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Cite This Source
Source: Boundless. “Binomial Probability Distributions.” Boundless Statistics. Boundless, 21 Jul. 2015. Retrieved 30 Nov. 2015 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/probabilityandvariability9/thebinomialrandomvariable37/binomialprobabilitydistributions1822630/