A department store manager has monitored the number of...

A department store manager has monitored the number of complaints received per week about poor service. The probabilities for numbers of complaints in a week, established by this review, are shown in the following table. Let $A$ be the event "there will be at least one complaint in a week" and $B$ the event "there will be fewer than ten complaints in a week." $$ \begin{array}{ccccccc} \hline \begin{array}{c} \text { Number of } \\ \text { complaints } \end{array} & 0 & 1 \text { to } 3 & 4 \text { to } 6 & 7 \text { to } 9 & 10 \text { to } 12 & \text { More } \\ \hline \text { Probability } & 0.14 & 0.39 & 0.23 & 0.15 & 0.06 & 0.03 \\ \hline \end{array} $$ a. Find the probability of $A$. b. Find the probability of $B$. c. Find the probability of the complement of $A$. d. Find the probability of the union of $A$ and $B$. e. Find the probability of the intersection of $A$ and $B$. f. Are $A$ and $B$ mutually exclusive? g. Are $A$ and $B$ collectively exhaustive?

A department store manager has monitored the number of complaints received per week about poor service. The probabilities for numbers of complaints in a week, established by this review, are shown in the following table. Let $A$ be the event "there will be at least one complaint in a week" and $B$ the event "there will be fewer than ten complaints in a week."
$$
\begin{array}{ccccccc}
\hline \begin{array}{c}
\text { Number of } \\
\text { complaints }
\end{array} & 0 & 1 \text { to } 3 & 4 \text { to } 6 & 7 \text { to } 9 & 10 \text { to } 12 & \text { More } \\
\hline \text { Probability } & 0.14 & 0.39 & 0.23 & 0.15 & 0.06 & 0.03 \\
\hline
\end{array}
$$
a. Find the probability of $A$.
b. Find the probability of $B$.
c. Find the probability of the complement of $A$.
d. Find the probability of the union of $A$ and $B$.
e. Find the probability of the intersection of $A$ and $B$.
f. Are $A$ and $B$ mutually exclusive?
g. Are $A$ and $B$ collectively exhaustive?

Key Concepts

Probability of an Event

This concept involves quantifying the likelihood that a specific outcome or event will occur. It is defined as a number between 0 and 1, where 0 signifies an impossible event and 1 signifies a certain event. In practical problems, probabilities are often given or calculated using a frequency or a theoretical model.

Complement Rule

The complement rule is a fundamental principle used in probability that states the probability of an event not occurring is equal to 1 minus the probability that it does occur. This rule is especially useful when it is simpler to calculate the probability of an event's complement rather than the event itself.

Union of Events

The union of events refers to the event that at least one of multiple events occurs. The probability of the union of two events is calculated by adding their individual probabilities and then subtracting the probability of their intersection to account for any overlap. This concept is critical when dealing with scenarios where events may not be mutually exclusive.

Intersection of Events

The intersection of events represents the scenario where two or more events occur simultaneously. In probability, the intersection is used to determine the likelihood that multiple conditions are met at the same time, playing a key role in problems where events might influence or overlap with each other.

Mutually Exclusive Events

Events are mutually exclusive if they cannot occur at the same time; that is, the occurrence of one event excludes the possibility of the other occurring. In probability problems, recognizing mutually exclusive events simplifies calculations because the probability of their intersection is zero.

Collectively Exhaustive Events

Collectively exhaustive events are a set of events that represent all possible outcomes of an experiment, meaning that at least one of the events must occur. This concept ensures that the total probability over the entire sample space sums to 1, covering every possible scenario.

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