A corporation takes delivery of some new machinery that must...

A corporation takes delivery of some new machinery that must be installed and checked before it becomes available to use. The corporation is sure that it will take no more than 7 days for this installation and check to take place. Let $A$ be the event "it will be more than 4 days before the machinery becomes available" and $B$ be the event "it will be less than 6 days before the machinery becomes available." a. Describe the event that is the complement of event $A$. b. Describe the event that is the intersection of events $A$ and $B$. c. Describe the event that is the union of events $A$ and $B$. d. Are events $A$ and $B$ mutually exclusive? e. Are events $A$ and $B$ collectively exhaustive? f. Show that $(A \cap B) \cup(\bar{A} \cap B)=B$. g. Show that $A \cup(\bar{A} \cap B)=A \cup B$.

A corporation takes delivery of some new machinery that must be installed and checked before it becomes available to use. The corporation is sure that it will take no more than 7 days for this installation and check to take place. Let $A$ be the event "it will be more than 4 days before the machinery becomes available" and $B$ be the event "it will be less than 6 days before the machinery becomes available."
a. Describe the event that is the complement of event $A$.
b. Describe the event that is the intersection of events $A$ and $B$.
c. Describe the event that is the union of events $A$ and $B$.
d. Are events $A$ and $B$ mutually exclusive?
e. Are events $A$ and $B$ collectively exhaustive?
f. Show that $(A \cap B) \cup(\bar{A} \cap B)=B$.
g. Show that $A \cup(\bar{A} \cap B)=A \cup B$.

Key Concepts

Collectively Exhaustive Events

Collectively exhaustive events are a set of events whose union covers the entire sample space, meaning that at least one of the events must occur. This concept ensures that no outcome in the sample space is left unaccounted for, which is essential for a comprehensive probability model.

Set Identities and Operations

Set identities, including those involving intersections, unions, and complements, are fundamental tools in both set theory and probability. They allow for the simplification and manipulation of complex events, provide a mathematical framework for understanding relationships between different events, and are used to prove equalities or identities such as distributive, associative, and De Morgan’s laws.

Mutually Exclusive Events

Mutually exclusive events are those events that cannot occur at the same time; that is, their intersection is an empty set. This concept implies that if one event occurs, the other cannot, which simplifies probability calculations since the probability of their union is just the sum of their individual probabilities.

Complement of an Event

The complement of an event A in probability theory refers to the set of all outcomes in the sample space that are not included in A. It is generally denoted as A? or A^c, and it plays a central role in calculating probabilities, especially when it is easier to determine the likelihood of 'not A' than A itself.

Intersection of Events

The intersection of events, denoted by A ? B, refers to the set of outcomes that are common to both events A and B. This concept is fundamental in probability theory for determining the probability that both events occur simultaneously.

Union of Events

The union of events, denoted by A ? B, represents the set of outcomes that occur in at least one of the events A or B. In other words, it includes any outcome that belongs to either A, B, or both, and is a basic construct when assessing the likelihood of one or more events happening.

Transcript

00:01 Right now in this problem we were told that a is the event that it will be more than four days before the machinery becomes available part a tells us to describe the complement of a the complement of a just means the exact opposite of a so if a is that it will take more than four days the complement be that it will take four days or less that would be the opposite of an a now on b we want to describe the event that is the intersection of a and b so a is that it will take more than four days and be easy that it will take less than six days.

00:46 So more than four and less than six means that it will take five days.

00:58 It's more than four and it's less than six.

01:01 It's got to take five days.

01:03 On c, we want to describe the union.

01:07 This means greater than four or less than six.

01:12 Greater than four and less than six just covers everything.

01:15 So this just means, i'm just going to say it will take a certain number of days.

01:29 It could take any amount of days.

01:30 More than four or less than four, since that covers everything.

01:34 And so it'll just take a certain number of days.

01:37 Now, indeed, we want to know, are they mutually exclusive? they are not mutually exclusive because a intersect b is not the empty set.

01:47 Okay, there are, as we saw there on b, it is possible for the intersection to occur, but it is collectively exhausted.