In a poll, 200 subscribers to Financial Magazine X indicated which of five specific companies they own stock in. The results are shown in the table above. If 15 of the 200 own stock in both IBM and AT & T, how many of those polled own stock in neither company?
Show Hint
For problems involving two overlapping groups, the formula \(\text{Total} = \text{Group 1} + \text{Group 2} - \text{Both} + \text{Neither}\) is a powerful shortcut. Here, \(200 = 30 + 48 - 15 + \text{Neither}\), which simplifies to \(200 = 63 + \text{Neither}\), giving \(\text{Neither} = 137\).
Step 1: Understanding the Concept:
This is a problem involving set theory. We need to find the number of elements that are outside of two given sets (people who own neither AT & T nor IBM stock) within a universal set (all 200 people polled). Step 2: Key Formula or Approach:
We can use the Principle of Inclusion-Exclusion. For two sets A and B, the total number of elements is given by:
\[ \text{Total} = |A \text{ only}| + |B \text{ only}| + |\text{Both}| + |\text{Neither}| \]
A more direct formula is:
\[ \text{Total} = |A \cup B| + |\text{Neither}| \]
where \(|A \cup B| = |A| + |B| - |A \cap B|\) is the number of people owning at least one of the stocks. Step 3: Detailed Explanation:
Let A be the set of people who own AT & T stock and B be the set of people who own IBM stock.
From the given data:
Total number of subscribers polled = 200.
Number who own AT & T stock, \(|A|\) = 30.
Number who own IBM stock, \(|B|\) = 48.
Number who own both, \(|A \cap B|\) = 15.
First, find the number of people who own at least one of the two stocks (\(|A \cup B|\)):
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
\[ |A \cup B| = 30 + 48 - 15 = 63 \]
This means 63 people own either AT & T stock, or IBM stock, or both.
Now, to find the number of people who own neither, we subtract this number from the total number of people polled:
\[ |\text{Neither}| = \text{Total} - |A \cup B| \]
\[ |\text{Neither}| = 200 - 63 = 137 \] Step 4: Final Answer:
137 people own stock in neither AT & T nor IBM. This corresponds to option (E).
