Consider n independent trials of an experiment in which each...

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 $C_{k} p^{k} q^{n-k}$ 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 $\left(\frac{1}{3}+\frac{2}{3}\right)^{8}$ .

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 $C_{k} p^{k} q^{n-k}$ 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 $\left(\frac{1}{3}+\frac{2}{3}\right)^{8}$ .

Key Concepts

Bernoulli Trial

A Bernoulli trial is an experiment that produces exactly two outcomes, typically labeled as 'success' and 'failure'. Each trial has a fixed probability of success (p) and a complementary probability of failure (1-p), and these probabilities remain constant over repeated trials.

Independent Trials

Independent trials refer to experiments in which the outcome of any one trial does not affect the outcomes of any other trials. This property is crucial in probabilistic models such as the binomial distribution where each trial's outcome is unaffected by the previous ones.

Binomial Experiment

A binomial experiment is a statistical experiment that consists of a fixed number of independent Bernoulli trials. Each trial has the same probability of success, and the experiment is often analyzed to determine the probability of achieving a certain number of successes in those trials.

Binomial Coefficient

The binomial coefficient, commonly written as 'n choose k', represents the number of ways to choose k successes from n independent trials. It is a key component in calculating probabilities in a binomial experiment as it accounts for all the possible arrangements of successes and failures.

Binomial Probability Distribution

The binomial probability distribution describes the probability of obtaining exactly k successes in n independent Bernoulli trials. It is given by the formula: (n choose k) multiplied by p^k and (1-p)^(n-k), where p is the probability of success. This distribution applies directly to situations involving repeated trials with two outcomes.

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