Probability In Exercises 85-88, consider n independent...

Probability In Exercises $85-88$, consider $n$ independent trials of an experiment in which each trial has two possible outcomes: "success" or "failure" The probability of a success on each trial is $p,$ and the probability of a failure is $q=1-p$. In this context, the term $_{n} C_{k} p^{k} q^{n-k}$ in the expansion of $(p+q)^{n}$ gives the probability of $k$ successes in the $n$ trials of the experiment. The probability of a sales representative making a sale with any one customer is $\frac{1}{3}$. The sales representative makes eight contacts a day. To find the probability of making four sales, evaluate the term $_{8} C_{4}\left(\frac{1}{3}\right)^{4}\left(\frac{2}{3}\right)^{4}$ in the expansion of $\left(\frac{1}{3}+\frac{2}{3}\right)^{8}$

Probability In Exercises $85-88$, consider $n$ independent trials of an experiment in which each trial has two possible outcomes: "success" or "failure" The probability of a success on each trial is $p,$ and the probability of a failure is $q=1-p$. In this context, the term $_{n} C_{k} p^{k} q^{n-k}$ in the expansion of $(p+q)^{n}$ gives the probability of $k$ successes in the $n$ trials of the experiment.
The probability of a sales representative making a sale with any one customer is $\frac{1}{3}$. The sales representative makes eight contacts a day. To find the probability of making four sales, evaluate the term $_{8} C_{4}\left(\frac{1}{3}\right)^{4}\left(\frac{2}{3}\right)^{4}$ in the expansion of $\left(\frac{1}{3}+\frac{2}{3}\right)^{8}$

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Key Concepts

Binomial Coefficient

A combinatorial term represented as C(n, k), computed as n!/(k!(n-k)!), which counts the number of distinct ways to select k successes from n trials. It is crucial in enumerating the possible outcomes in a binomial experiment.

Bernoulli Trial

A single experiment or trial that results in one of two outcomes, typically labeled as success or failure. Bernoulli trials are independent with a constant probability of success and form the basis of the binomial distribution.

Binomial Distribution

A probability distribution that describes the number of successes in a fixed number of independent trials, each with two possible outcomes (success or failure) and a constant probability of success. It is foundational in assessing probabilities in binary outcome experiments.

Binomial Probability Formula

An expression given by C(n, k) * p^k * q^(n-k) that calculates the probability of obtaining exactly k successes in n independent trials, where p is the probability of success and q = 1-p is the probability of failure.

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