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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$q=1p$ .  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
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
Since the trials are independent and since the probabilities of success and failure on each trial are, respectively,
These probabilities are called binomial probabilities, and the random variable
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
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Key Term Reference
 binomial distribution
 Appears in these related concepts: The Hypergeometric Random Variable, Calculating a Normal Approximation, and Comparing Two Populations: Paired Difference 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: Probability Histograms, The Rise of Independents, and Unions and Intersections
 mutually exclusive
 Appears in these related concepts: Complementary Events, Multiple IRRs, and Solving Systems of Linear Inequalities
 probability
 Appears in these related concepts: The Addition Rule, Theoretical Probability, and Rules of Probability for Mendelian Inheritance
 probability distribution
 Appears in these related concepts: Probability Distributions for Discrete Random Variables, Recognizing and Using a Histogram, 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, Controlling for a Variable, and Math Review
Sources
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Cite This Source
Source: Boundless. “Binomial Probability Distributions.” Boundless Statistics. Boundless, 20 May. 2016. Retrieved 26 May. 2016 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/probabilityandvariability9/thebinomialrandomvariable37/binomialprobabilitydistributions1822630/