A production manager knows that 5 % of components produced...

A production manager knows that $5 \%$ of components produced by a particular manufacturing process have some defect. Six of these components, whose characteristics can be assumed to be independent of each other, are examined. a. What is the probability that none of these components has a defect? b. What is the probability that one of these components has a defect? c. What is the probability that at least two of these components have a defect?

A production manager knows that $5 \%$ of components produced by a particular manufacturing process have some defect. Six of these components, whose characteristics can be assumed to be independent of each other, are examined.
a. What is the probability that none of these components has a defect?
b. What is the probability that one of these components has a defect?
c. What is the probability that at least two of these components have a defect?

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Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution used to represent the number of successes in a fixed number of independent trials, each with an identical probability of success. It is appropriate for situations where each trial has only two outcomes (such as 'defective' or 'non-defective') and the outcomes of the trials are independent.

Independent Events

Independent events are those where the occurrence of one event does not influence the occurrence of another. This concept is crucial when computing probabilities for multiple trials, as it enables the multiplication of probabilities of individual events to determine the joint probability.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability of it not occurring. This rule is especially useful for calculating probabilities of events like 'at least' scenarios, by subtracting the probability of the undesired outcomes from one.

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