A corporation receives a particular part in shipments of 100...

A corporation receives a particular part in shipments of 100 . Research indicated the probabilities shown in the accompanying table for numbers of defective parts in a shipment. $$ \begin{array}{lccccc} \hline \text { Number } & 0 & 1 & 2 & 3 & >3 \text { defective } \\ \hline \text { Probability } & 0.29 & 0.36 & 0.22 & 0.10 & 0.03 \\ \hline \end{array} $$ a. What is the probability that there will be fewer than three defective parts in a shipment? b. What is the probability that there will be more than one defective part in a shipment? c. The five probabilities in the table sum to 1. Why must this be so?

A corporation receives a particular part in shipments of 100 . Research indicated the probabilities shown in the accompanying table for numbers of defective parts in a shipment.
$$
\begin{array}{lccccc}
\hline \text { Number } & 0 & 1 & 2 & 3 & >3 \text { defective } \\
\hline \text { Probability } & 0.29 & 0.36 & 0.22 & 0.10 & 0.03 \\
\hline
\end{array}
$$
a. What is the probability that there will be fewer than three defective parts in a shipment?
b. What is the probability that there will be more than one defective part in a shipment?
c. The five probabilities in the table sum to 1. Why must this be so?

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Key Concepts

Discrete Probability Distribution

A discrete probability distribution assigns a probability to each individual outcome in a finite set of outcomes. In statistical problems, such as evaluating defective parts in shipments, each specific count of defects is paired with a probability, constructing the entire distribution for the random variable of interest.

Cumulative Probability

Cumulative probability involves summing the probabilities of several outcomes to obtain the probability that a variable takes a value within a certain range. This concept is used when finding probabilities for events like having fewer than a given number of defects or more than a given threshold by adding up the relevant individual probabilities.

Exhaustive Events and Total Probability

The requirement that the probabilities must sum to 1 reflects the principle of exhaustive events in probability. This principle states that all possible outcomes of an experiment must be accounted for in the distribution, ensuring that one of the outcomes must occur, hence the total probability equals 1.

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