Problem 14 - A First Course in Probability

Question

The following data were given in a study of a group of 1000 subscribers to a certain magazine: In reference to job, marital status, and education, there were 312 professionals, 470 married persons, 525 college graduates, 42 professional college graduates, 147 married college graduates, 86 married professionals, and 25 married professional college graduates. Show that the numbers reported in the study must be incorrect. Hint: Let \(M, W,\) and \(G\) denote, respectively, the set of professionals, married persons, and college graduates. Assume that one of the 1000 persons is chosen at random, and use Proposition 4.4 to show that if the given numbers are correct, then \(P(M \cup W \cup G)>1\).

Step-by-Step Solution

Answer

The given data results in a probability greater than 1 when applying the Principle of Inclusion-Exclusion: \(P(M \cup W \cup G) = \frac{557}{500}\). Since this is not possible, the reported numbers in the study must be incorrect.

Step 1: Calculate the Probabilities of the Sets

Calculate the probabilities for each set: - \(P(M) = \frac{312}{1000}\) - \(P(W) = \frac{470}{1000}\) - \(P(G) = \frac{525}{1000}\) - \(P(M \cap G) = \frac{42}{1000}\) - \(P(M \cap W) = \frac{86}{1000}\) - \(P(W \cap G) = \frac{147}{1000}\) - \(P(M \cap W \cap G) = \frac{25}{1000}\)

Step 2: Apply the Principle of Inclusion-Exclusion

Using the Principle of Inclusion-Exclusion, compute the probability of the union of the three sets: \(P(M \cup W \cup G) = P(M) + P(W) + P(G) - P(M \cap W) - P(M \cap G) - P(W \cap G) +P(M \cap W \cap G)\)

Step 3: Plug in the Probabilities

Insert the calculated probabilities: \(P(M \cup W \cup G) = \frac{312}{1000} + \frac{470}{1000} + \frac{525}{1000} - \frac{86}{1000} - \frac{42}{1000} - \frac{147}{1000} + \frac{25}{1000}\)

Step 4: Evaluate the Probability

Calculate the probability of the union: \(P(M \cup W \cup G) = \frac{557}{500}\)

Step 5: Determine a Contradiction

Observe that \(P(M \cup W \cup G) >1\), which is not possible for any probability. This means that the reported numbers in the study must be incorrect.

Key Concepts

Probability Theory

Probability theory is the branch of mathematics that deals with modeling uncertainty. It helps us determine the likelihood of different outcomes. In simple terms, if you want to know how probable an event is, you use probability theory. In any probability problem,

• The probability of an event (such as flipping a heads on a coin toss) is a number between 0 and 1.
• This number reflects how likely the event will happen, with 0 meaning impossible and 1 meaning certain.

In our exercise, we calculate the probability of people belonging to certain groups, such as professionals or married people, out of a total of 1000 individuals. By adding or subtracting these probabilities, we can explore more complex queries about our group of people, such as how many people belong to at least one of the categories.

Set Operations

Set operations are a fundamental part of mathematical logic and probability. They describe how different groups or sets interact and overlap. Three key operations are:

• Union (\( \cup \)): Represents elements that belong to at least one of the sets. It’s like combining the items in two piles – you get everything from both.
• Intersection (\( \cap \)): Consists of elements common to both sets. Think of it as the overlap in a Venn diagram.
• Complement: Represents elements not in a particular set, which is like everything outside your pile.

In the exercise, we use these operations to determine the set of individuals who are professionals, married, or college graduates. By applying the principle of inclusion-exclusion, we know that simply adding the probabilities of separate groups overestimates the problem. To correct this, we subtract the probabilities of overlaps and further adjust using intersections.

Contradiction in Probability

A contradiction in probability arises when a calculated probability doesn’t make sense—such as getting a number greater than one. A probability greater than 1 is nonsensical because it suggests more certainty than "absolutely certain," which isn't possible. In the provided exercise, the use of the inclusion-exclusion principle led to such a situation.

• We calculated the probability for at least one of three sets (professionals, married, college graduates) and ended with \(P(M \cup W \cup G) = \frac{557}{500}\), or 1.114.
• Since probabilities must be between 0 and 1, this tells us something went wrong—most likely in the data given in the study.

Such contradictions highlight the importance of careful probability checks and can signal issues in measurement or data collection, prompting a need to review and correct underlying numbers or assumptions.