A company installs new central-heating furnaces and has...

A company installs new central-heating furnaces and has found that for $15 \%$ of all installations, a return visit is needed to make some modifications. Six installations were made in a particular week. Assume independence of outcomes for these installations. a. What is the probability that a return visit will be needed in all these cases? b. What is the probability that a return visit will be needed in none of these cases? c. What is the probability that a return visit will be needed in more than 1 of these cases?

A company installs new central-heating furnaces and has found that for $15 \%$ of all installations, a return visit is needed to make some modifications. Six installations were made in a particular week. Assume independence of outcomes for these installations.
a. What is the probability that a return visit will be needed in all these cases?
b. What is the probability that a return visit will be needed in none of these cases?
c. What is the probability that a return visit will be needed in more than 1 of these cases?

Key Concepts

Summation of Probabilities (Cumulative Distribution)

When considering scenarios that involve ranges of outcomes, such as having more than one return visit, it often requires summing the individual binomial probabilities of all relevant outcomes to achieve the total probability for the desired event.

Complementary Probability

Complementary probability involves calculating the probability of an event not occurring (or the complement of the desired event) to simplify the calculation of complex probabilities. This method is often used when it is easier to compute the probability of the complement than the event directly.

Independence of Trials

This concept stipulates that the outcome of one installation does not affect that of another, a key requirement for applying the binomial distribution. Independence ensures that the probability of multiple events occurring is the product of their individual probabilities.

Bernoulli Trials

Each individual installation is an example of a Bernoulli trial, which is a random experiment with exactly two possible outcomes: success or failure. The constant probability of success applies to each installation independently.

Binomial Distribution

The binomial distribution describes the probability of obtaining a fixed number of successes in a specific number of independent Bernoulli trials, each with the same success probability. It is frequently used to model scenarios where each trial can result in success (e.g. a return visit) or failure (no return visit).

Transcript

Please give Ace some feedback