Notes

Creating a Venn diagram to visually represent the distribution and relationship between three types of decorations (flowers, candles, and balloons) is a great way to explore various probabilities. While I can't draw directly here, I'll describe how to construct and interpret a Venn diagram for this scenario, and then discuss how to calculate the probabilities you're interested in.

Constructing the Venn Diagram:
Draw three overlapping circles, each representing one type of decoration: one for Flowers (F), one for Balloons (B), and one for Candles (C).
Intersection areas:
Where F and B overlap, indicate tables with both flowers and balloons.
Where B and C overlap, show tables with both balloons and candles.
Where F and C overlap, denote tables with both flowers and candles.
The center area, where all three circles overlap, represents tables with all three types of decorations.
Outside the circles, but within a rectangle enclosing the circles (representing the universal set of all tables), indicate tables without any of these three decorations (if applicable).
Example Scenario:
For illustration, let's assign some numbers to these categories (note: these numbers are hypothetical for explanation purposes):

Flowers (F): 15 tables
Balloons (B): 12 tables
Candles (C): 10 tables
Flowers and Balloons (F ∩ B): 5 tables
Balloons and Candles (B ∩ C): 4 tables
Flowers and Candles (F ∩ C): 6 tables
All three (F ∩ B ∩ C): 2 tables
Total tables: 20
Calculating Probabilities:
Probability a table contains balloons (P(B)):
To find
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P(B), count the total number of tables with balloons (including those with just balloons, those with balloons and one other decoration, and those with all three decorations), then divide by the total number of tables.

Probability a table has balloons given that it also has flowers (P(B|F)):
This is a conditional probability, which requires us to focus only on the tables with flowers, then find the proportion of those that also have balloons. The formula is:

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P(B∣F)=
P(F)
P(B∩F)
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Where:

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P(B∩F) is the probability of a table having both balloons and flowers.
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P(F) is the probability of a table having flowers.
Proper Mathematical Notation and Terminology:
∩: Intersection, representing tables that have both/all of the specified decorations.
|: Given that, used in conditional probability to explore the probability of one event given another.
P(Event): Probability of the event.
Example Calculation:

If we want to calculate the probability a table contains balloons (
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P(B)), assuming the hypothetical numbers above and proper distribution, we'd sum all instances involving balloons divided by the total tables.

Since I can't draw the Venn diagram here, let's focus on the mathematical aspect and calculate the provided probability example with the hypothetical values. For a more specific calculation or further exploration of probabilities within this setup, more detailed distribution information would be necessary.