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A two-way table is a valuable tool that helps organize and display the information regarding the associations between two categorical variables. In the context of our problem, these variables are students' usage of Facebook and MySpace. By translating the probabilities of certain events into numbers of students, we can break down the total into categories such as both, neither, only Facebook, and only MySpace.

For instance, considering there are 20 million college students, and knowing the percentages:

• 85% have profiles on Facebook (\( P(F) = 0.85 \)).
• 54% use MySpace regularly (\( P(M) = 0.54 \)).
• 42% use both platforms (\( P(F \cap M) = 0.42 \)).

We can calculate:

• 8.4 million students use both (0.42 * 20 million).
• 8.6 million only use Facebook (0.85 * 20 million - 8.4 million).
• 2.2 million only use MySpace (0.54 * 20 million - 8.4 million).
• 0.8 million use neither (20 - (8.6 + 2.2 + 8.4) million).

The two-way table organizes these results, clearly showing who uses which platform, enabling a better visualization and understanding of the data.

The Venn diagram is an effective graphical representation for organizing and visualizing the overlaps and exclusivities between different sets or groups. In our exercise, it helps represent the groups of students using Facebook, MySpace, both, or neither.

For this, you draw two overlapping circles:

• One circle represents Facebook users (\( F \)).
• The other represents MySpace users (\( M \)).
• The overlapping section represents users of both platforms (\( F \cap M \)).

In the diagram:

• The intersection region covering 8.4 million students symbolizes those using both Facebook and MySpace.
• The remaining section within the Facebook circle, excluding the overlap, represents 8.6 million students who use only Facebook.
• Similarly, the MySpace circle's non-overlapping part accounts for the 2.2 million students using only MySpace.
• Additionally, the space outside the circles accounts for the 0.8 million students who neither use Facebook nor MySpace.

This visual aid helps simplify complex problems, making it easier to grasp relationships between different sets.

Symbolic form in probability allows us to express events using symbols and logical operations, making complex statements more concise and precise. For example, in our exercise, we look at the event where a student has posted a profile on at least one of the two sites, Facebook or MySpace.

Symbolically, the union of these events can be expressed as:

• \( F \cup M \)

This represents all students who have profiles on either Facebook, MySpace, or both. To find the probability of this event occurring, we use the formula:

• \( P(F \cup M) = P(F) + P(M) - P(F \cap M) \)

By substituting the given probabilities, we have:

• \( 0.85 + 0.54 - 0.42 = 0.97 \)

Thus, 97% is the probability of a student having an account on at least one platform. This method of using symbols and operations simplifies the calculations and presentation of solution.