Math Problem Statement

A survey of grocery stores in Eastern Canada revealed that 40% had a pharmacy, 50% has a floral shop, and 70% had a deli. Suppose 10% of the stores have all three departments, 30% have both a pharmany and a deli, 25% have both a floral shop and deli, and 20% have both a pharmacy and a floral shop.What is the name given to the event of "selecting a store with a pharmany, a floral shop, and a deli"?What is the probability of selecting a store that does not have all three departments?

Solution

In this context:

  1. The event of "selecting a store with a pharmacy, a floral shop, and a deli" is called the intersection of three events or joint occurrence of these three departments.

  2. To determine the probability of selecting a store that does not have all three departments, we can use the given probability information directly.

Since 10% of the stores have all three departments, the probability of selecting a store with all three departments is: P(PharmacyFloralDeli)=0.10P(\text{Pharmacy} \cap \text{Floral} \cap \text{Deli}) = 0.10

Therefore, the probability of selecting a store that does not have all three departments is the complement of this probability: P(not all three departments)=1P(all three departments)=10.10=0.90P(\text{not all three departments}) = 1 - P(\text{all three departments}) = 1 - 0.10 = 0.90

Answer Summary:

  1. The name for the event of selecting a store with a pharmacy, floral shop, and deli is the intersection of three events.
  2. The probability of selecting a store that does not have all three departments is 0.90 (or 90%).

Would you like more details on the solution or clarification? Here are some related questions to expand understanding:

  1. What is the probability of selecting a store that has at least one of these departments?
  2. How would you calculate the probability of a store having only two of the three departments?
  3. Can you verify the information given using a Venn diagram or inclusion-exclusion principle?
  4. How do you interpret joint probabilities in survey data?
  5. What would change in calculations if the survey included more departments?

Tip: For multi-category probability problems, drawing a Venn diagram helps visualize overlaps and interactions.

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Math Problem Analysis

Mathematical Concepts

Probability
Set Theory
Complementary Events

Formulas

P(A ∩ B ∩ C) = Probability of intersection of three events
P(not A ∩ B ∩ C) = 1 - P(A ∩ B ∩ C)

Theorems

Inclusion-Exclusion Principle

Suitable Grade Level

Grades 10-12