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Difficulty: MediumBasic Probability and Counting Methods

A community art center offers 4040 different workshops during a summer session. Among these workshops, 2222 are scheduled in the evening, 1818 are scheduled on weekends, and 88 are scheduled in both the evening and on weekends. If one workshop is selected at random from the 4040 workshops, what is the probability that it is scheduled in the evening, on a weekend, or both?

  1. A
    15\frac{1}{5}
  2. B
    310\frac{3}{10}
  3. C
    35\frac{3}{5}
  4. 45\frac{4}{5}Answer
  5. E
    11

Answer

The correct probability is 45\frac{4}{5}.
To find the probability that a randomly chosen workshop is in the evening, on a weekend, or both, calculate the total number of distinct workshops meeting at least one criterion using the formula N(Evening∪Weekend)=N(Evening)+N(Weekend)−N(Evening∩Weekend)N(\text{Evening} \cup \text{Weekend}) = N(\text{Evening}) + N(\text{Weekend}) - N(\text{Evening} \cap \text{Weekend}). Substituting the given values gives 22+18−8=3222 + 18 - 8 = 32 workshops. Dividing by the total 4040 workshops gives 3240\frac{32}{40}, which reduces to 45\frac{4}{5}.

Step-by-Step Solution

1
Identify the given counts for each category
Total workshops = 4040, Evening workshops = 2222, Weekend workshops = 1818, Both = 88.
Extract the necessary components to apply the inclusion-exclusion principle.
2
Calculate the number of workshops in the evening, on a weekend, or both
Number of favorable workshops = 22+18−8=3222 + 18 - 8 = 32.
Workshops scheduled in both categories are counted twice if evening and weekend counts are added directly, so the intersection must be subtracted once.
3
Compute the probability and simplify the fraction
Probability = 3240=45\frac{32}{40} = \frac{4}{5}.
Divide the number of favorable outcomes by the total sample space size.

Key Concept

Probability of Combined Events (Principle of Inclusion-Exclusion)
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Topics

Basic Probability and Counting Methods

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