A corporation has just received new machinery that must be...

A corporation has just received new machinery that must be installed and checked before it becomes operational. The accompanying table shows a manager's probability assessment for the number of days required before the machinery becomes operational. $$ \begin{array}{lrrrrr} \hline \text { Number of days } & 3 & 4 & 5 & 6 & 7 \\ \hline \text { Probability } & 0.08 & 0.24 & 0.41 & 0.20 & 0.07 \\ \hline \end{array} $$ Let $A$ be the event "it will be more than four days before the machinery becomes operational," and let $B$ be the event "it will be less than six days before the machinery becomes available." a. Find the probability of event $A$. b. Find the probability of event $B$. c. Find the probability of the complement of event $A$. d. Find the probability of the intersection of events $A$ and $B$. e. Find the probability of the union of events $A$ and $B$.

A corporation has just received new machinery that must be installed and checked before it becomes operational. The accompanying table shows a manager's probability assessment for the number of days required before the machinery becomes operational.
$$
\begin{array}{lrrrrr}
\hline \text { Number of days } & 3 & 4 & 5 & 6 & 7 \\
\hline \text { Probability } & 0.08 & 0.24 & 0.41 & 0.20 & 0.07 \\
\hline
\end{array}
$$
Let $A$ be the event "it will be more than four days before the machinery becomes operational," and let $B$ be the event "it will be less than six days before the machinery becomes available."
a. Find the probability of event $A$.
b. Find the probability of event $B$.
c. Find the probability of the complement of event $A$.
d. Find the probability of the intersection of events $A$ and $B$.
e. Find the probability of the union of events $A$ and $B$.

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Key Concepts

Intersection of Events

The intersection of two events contains the outcomes that are common to both events. The probability of the intersection is the likelihood that both events occur simultaneously.

Discrete Probability Distribution

A discrete probability distribution assigns a probability to each possible outcome in a finite or countable sample space. It ensures that the total probability over all outcomes equals 1 and is used when outcomes can be precisely listed, as in the days required for machinery installation.

Event

An event is a set of outcomes from a probability experiment. It can be a single outcome or a collection of outcomes, and its probability reflects the likelihood that any of those outcomes occurs.

Complement of an Event

The complement of an event consists of all outcomes in the sample space that are not included in the event. The probability of the complement event is calculated as 1 minus the probability of the original event.

Union of Events

The union of two events is the set that contains all outcomes that are in at least one of the events. To calculate its probability accurately without double counting overlapping outcomes, the formula P(A) + P(B) - P(A ? B) is used.

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