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The report "Twitter in Higher Education: Usage Habits and Trends of Today's College Faculty" (Magna Publications, September 2009) describes results of a survey of nearly 2000 college faculty. The report indicates the following: \- \(30.7 \%\) reported that they use Twitter and \(69.3 \%\) said that they did not use Twitter. \- Of those who use Twitter, \(39.9 \%\) said they sometimes use Twitter to communicate with students. \- Of those who use Twitter, \(27.5 \%\) said that they sometimes use Twitter as a learning tool in the classroom. Consider the chance experiment that selects one of the study participants at random and define the following events: \(T=\) event that selected faculty member uses Twitter \(C=\) event that selected faculty member sometimes uses Twitter to communicate with students \(L=\) event that selected faculty member sometimes uses Twitter as a learning tool in the classroom a. Use the given information to determine the following probabilities: i. \(\quad P(T)\) ii. \(P\left(T^{C}\right)\) iii. \(P(C \mid T)\) iv. \(P(L \mid T)\) v. \(P(C \cap T)\) b. Interpret each of the probabilities computed in Part (a). c. What proportion of the faculty surveyed sometimes use Twitter to communicate with students? [Hint: Use the law of total probability to find \(P(C) .]\) d. What proportion of faculty surveyed sometimes use Twitter as a learning tool in the classroom?

Step by step solution

01

Calculate probabilities of individual events

Using the information from the survey, you can determine that: \n- \(P(T) = 0.307\) (This is given by the percentage of faculty that use Twitter), - \(P(T^{C}) = 1 - P(T) = 0.693\) (This is calculated based on the fact that either a person uses Twitter or does not)

02

Calculate conditional probabilities

Also given are: - \(P(C|T) = 0.399\) (This is the percentage of Twitter users who use it to communicate with students), - \(P(L|T) = 0.275\) (This is the percentage of Twitter users who use it as a learning tool)

03

Calculate the probability of the intersection of events

The intersection of two events (in this case, Twitter usage and communication with students) is given by: \(P(C \cap T) = P(T) \cdot P(C|T) = 0.307 \cdot 0.399 = 0.122493\) (rounded to six decimal places)

04

Compute probabilities using law of total probability

The law of total probability can be used to find the probability of an event by considering all the ways that it could happen:- \(P(C) = P(T) \cdot P(C|T) + P(T^C) \cdot P(C|T^C)\). Because we don't know \(P(C|T^C)\), we'll assume that if the faculty member does not use twitter, he/she cannot communicate via twitter. Therefore, \(P(C|T^C) = 0\). Then the equation becomes: \(P(C) = 0.307 \cdot 0.399 = 0.122493\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory

Probability theory is a branch of mathematics focused on analyzing random events. The fundamental idea is to quantify the likelihood of different outcomes—in this case, the behaviors of college faculty regarding Twitter use. Understanding probability helps us to make informed predictions or decisions when faced with uncertainty.

Key terms include the probability of an event, which is a measure from 0 to 1 indicating how likely the event is to occur, with 0 being impossible and 1 being certain. In our exercise, the event 'T', which signifies using Twitter, has a probability of 0.307, meaning it is a likely but not guaranteed event for a randomly selected faculty member.

Another vital term is the sample space, representing all possible outcomes. Here, the sample space includes all faculty members surveyed, dividing them into users and non-users of Twitter. By analyzing these probabilities through practical examples, students can develop a keen understanding of how theoretical probability applies to real-world scenarios.

Conditional Probability

Conditional probability is the probability of an event occurring, given that another event has already occurred. Represented as (P(A|B)), it reads as 'the probability of event A given event B.'

For instance, the exercise asks for the probability that a faculty member uses Twitter to communicate with students, given that they already use Twitter, denoted as (P(C|T)). This is given by 0.399, which means almost 40% of Twitter-using faculty communicate with students through the platform. Grasping conditional probability is crucial for interpreting data where certain conditions or criteria are focused on.

Calculating Conditional Probability

In calculations, the formula often used is: (P(A|B) = \[\frac{P(A \bigcap B)}{P(B)}\]). In terms of the survey, we think of the probability of two events—using Twitter and communicating with students—both happening, and then we divide by the probability of the given condition (using Twitter).

Law of Total Probability

The law of total probability allows us to calculate the probability of an event based on the probabilities of a set of related events. It's particularly effective when dealing with conditional probabilities. In the presented exercise, it's used to determine the proportion of faculty who use Twitter for communication, considering both users and non-users of Twitter.

To find the overall probability (P(C)) of a faculty member using Twitter to communicate, we sum the probability of communicating while being a Twitter user (P(C \bigcap T)) and while not being a Twitter user (P(C \bigcap T^C)). Since faculty cannot communicate via Twitter if they don't use it, (P(C|T^C)) is 0, and the equation simplifies. Applying the law of total probability simplifies complex problems into manageable parts and is a powerful tool for comprehensive statistical analysis.

Application of the Law

The formula looks like this: (P(C) = P(T) \bigcdot \[\frac{P(C \bigcap T)}{P(T)}\] + P(T^C) \bigcdot \[\frac{P(C \bigcap T^C)}{P(T^C)}\]). Since the second part of the equation is zero, we only consider the first part to find the overall probability of faculty using Twitter for student communication.