The probability of success on a single trial of a binomial...

The probability of success on a single trial of a binomial experiment is known to be $\frac{1}{4} .$ The random variable $x$, number of successes, has a mean value of $80 .$ Find the number of trials involved in this experiment and the standard deviation of $x.$

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Key Concepts

Binomial Distribution

A binomial distribution models the number of successes in a fixed number of independent trials, each having two outcomes: success or failure. It is defined by two parameters: the number of trials (n) and the probability of success (p). This concept is fundamental in probability theory for processes with a clear binary outcome executing under identical conditions.

Mean of a Binomial Distribution

The mean (or expected value) of a binomial distribution is calculated as the product of the number of trials (n) and the probability of success (p). This measure represents the average number of successes one can expect over many repetitions of the experiment, providing insight into the central tendency of the distribution.

Standard Deviation of a Binomial Distribution

The standard deviation quantifies the dispersion or spread in the number of successes around the mean in a binomial experiment. It is computed by taking the square root of the variance, where the variance is given by n * p * (1 - p). This calculation provides a sense of the variability inherent in the process of repeated trials.

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