A market research survey finds that 50% of consumers prefer product A, 40% prefer product B, and 20% prefer product C. Additionally, 15% prefer both A and B, 10% prefer both A and... A market research survey finds that 50% of consumers prefer product A, 40% prefer product B, and 20% prefer product C. Additionally, 15% prefer both A and B, 10% prefer both A and C, and 5% prefer all three products. If 60% of consumers prefer none of the products, what percentage of consumers prefer both B and C?
Understand the Problem
This is a probability problem involving overlapping sets of consumer preferences for products A, B, and C. We're given the individual preferences, the overlap between A and B, A and C, and all three. We also know the percentage who prefer none of the products. The goal is to find the percentage of consumers who prefer both B and C.
Answer
$7\%$
Answer for screen readers
$7%$
Steps to Solve
- Define variables
Let $P(A)$, $P(B)$, and $P(C)$ represent the percentage of consumers who prefer products A, B, and C, respectively.
Let $P(A \cap B)$, $P(A \cap C)$, and $P(B \cap C)$ represent the percentage of consumers who prefer both A and B, A and C, and B and C, respectively.
Let $P(A \cap B \cap C)$ represent the percentage of consumers who prefer all three products.
Let $P(\text{None})$ represent the percentage of consumers who prefer none of the products.
- Write down the given information
$P(A) = 42%$
$P(B) = 31%$
$P(C) = 28%$
$P(A \cap B) = 8%$
$P(A \cap C) = 5%$
$P(A \cap B \cap C) = 3%$
$P(\text{None}) = 26%$
- Use the principle of inclusion-exclusion to find the percentage of consumers who prefer at least one product
The principle of inclusion-exclusion states that: $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$
Since $26%$ prefer none, it follows that $100% - 26% = 74%$ prefer at least one product. Thus, $P(A \cup B \cup C) = 74%$
- Substitute the given values into the inclusion-exclusion formula
$74 = 42 + 31 + 28 - 8 - 5 - P(B \cap C) + 3$
- Solve for $P(B \cap C)$
Combine the numbers: $74 = 91 - 13 - P(B \cap C) + 3$
$74 = 78 + 3 - P(B \cap C)$
$74 = 81 - P(B \cap C)$
$P(B \cap C) = 81 - 74$
$P(B \cap C) = 7$
Therefore, the percentage of consumers who prefer both B and C is $7%$.
$7%$
More Information
The inclusion-exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two sets: $|A \cup B| = |A| + |B| - |A \cap B|$. For three sets, the principle is: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$.
Tips
A common mistake involves misapplying the inclusion-exclusion principle or making arithmetic errors when substituting values into the formula and solving for the unknown. Carefully double-checking the calculations and ensuring that the correct terms are included with the correct signs can help avoid these errors. Another mistake is to forget to account for the percentage of consumers that prefer none of the products when determining $P(A \cup B \cup C)$.
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