91Ó°ÊÓ

A recent survey reported in Bloomberg Businessweek dealt with the salaries of CEOs at large corporations and whether company shareholders made money or lost money. $$ \begin{array}{|cccc|} \hline & \begin{array}{c} \text { CEO Paid More } \\ \text { Than \$1 Million } \end{array} & \begin{array}{c} \text { CEO Paid Less } \\ \text { Than \$1 Million } \end{array} & \text { Total } \\ \hline \text { Shareholders made money } & 2 & 11 & 13 \\ \text { Shareholders lost money } & \underline{4} & 3 & \frac{7}{20} \\ \hline \text { Total } & 6 & 14 & 20 \\ \hline \end{array} $$ If a company is randomly selected from the list of 20 studied, what is the probability: a. The CEO made more than \(\$ 1\) million? b. The CEO made more than \(\$ 1\) million or the shareholders lost money? c. The CEO made more than \(\$ 1\) million given the shareholders lost money? d. Of selecting two CEOs and finding they both made more than \(\$ 1\) million?

Step by step solution

01

Understanding the Given Table and Probabilities

The table summarizes the number of corporations where a CEO made more or less than \$1 million and whether shareholders made or lost money. Before solving each part, recognize that our total sample size is 20. For any event probability, we use the formula: \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \).

02

Calculating Probability for Part (a)

For part (a), find the probability that the CEO made more than \\(1 million. This is simply the total number of companies where CEOs received more than \\)1 million over the total number of companies. We can see there are 6 such companies. Hence, \( P(\text{CEO paid > \$1 million}) = \frac{6}{20} = 0.3 \).

03

Calculating Probability for Part (b)

For part (b), we calculate the probability that the CEO made more than \\(1 million or shareholders lost money. We use the principle of inclusion-exclusion: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). We have \( P(A) = \frac{6}{20} \), \( P(B) = \frac{7}{20} \), and \( P(A \cap B) = \frac{4}{20} \) (both CEO made > \\)1 million and shareholders lost money). Thus, \( P(\text{CEO > \$1M or Shareholders lost}) = \frac{6}{20} + \frac{7}{20} - \frac{4}{20} = \frac{9}{20} = 0.45 \).

04

Calculating Probability for Part (c)

For part (c), find the probability that the CEO made more than \\(1 million given that shareholders lost money. This is a conditional probability problem, given by \( P(A | B) = \frac{P(A \cap B)}{P(B)} \). Here, \( A \cap B \) is 4, and \( B \) is 7 (from the table). Therefore, \( P(\text{CEO > \\)1M | Shareholders lost}) = \frac{4}{7} \approx 0.571 \).

05

Calculating Probability for Part (d)

In part (d), we need the probability both selected CEOs made more than \$1 million. Assume the selections are independent and without replacement. The probability for the first CEO is \( \frac{6}{20} \), and for the second CEO is \( \frac{5}{19} \) (one less CEO and total). Thus, the probability is \( \frac{6}{20} \times \frac{5}{19} = \frac{30}{380} \approx 0.079 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability

Conditional probability helps us find the probability of an event occurring, given that another event has already happened. This is especially useful when the occurrence of one event affects the likelihood of another. For example, in the CEO salary exercise, the probability that a CEO made more than \( \\(1 \) million is needed under the condition that shareholders lost money. To calculate conditional probability, use the formula: \[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]- \( P(A | B) \) is the probability of event A occurring given that B has occurred. - \( P(A \cap B) \) is the probability that both A and B occur together. - \( P(B) \) is the probability that event B occurs.In our exercise, event A is the CEO making more than \( \\)1 \) million, and event B is shareholders losing money. By calculating these probabilities, you can determine how likely it is for the CEO to earn higher when shareholders have a loss.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a useful tool in probability theory to handle situations where you need to calculate the probability of either of two events occurring. Simply adding the probabilities of both events might count some overlap twice, so this principle helps correct that. In the context of our CEO salary and shareholder outcome problem, it is used to find the probability that either the CEO made more than \( \$1 \) million or the shareholders lost money.The formula is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]- \( P(A \cup B) \) represents the probability of either event A or B occurring.- \( P(A) \) and \( P(B) \) are the individual probabilities of events A and B, respectively. - \( P(A \cap B) \) is the probability that both events occur together, and it is subtracted because it is counted in both \( P(A) \) and \( P(B) \).

Sample Space

In probability theory, the sample space is the set of all possible outcomes of an experiment. Knowing the sample space is the first step in calculating probabilities because it gives you the total number of possible outcomes to consider. In the given exercise, the sample space consists of the 20 companies surveyed about CEO salaries and shareholder profit or loss.Key features of a sample space include:

• Having all possible outcomes of an event.
• Outcomes should be mutually exclusive (no overlap).
• Outcomes must be exhaustive (covering all possibilities).

With the sample space understood, probabilities of different events can be calculated, such as the probability of a CEO making more or less than \( \$1 \) million.

Event Probability

Event probability refers to the likelihood of a specific outcome or set of outcomes occurring within a given sample space. In our exercise, different events are considered, such as CEOs earning more than \( \$1 \) million or shareholders losing money.To calculate the probability of an event, use the general formula: \[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]Important points to consider when calculating event probability include:

• Identify the total number of outcomes in the sample space.
• Determine the number of outcomes that correspond to the event of interest.
• Divide the number of favorable outcomes by the total number of outcomes to find the probability.

Detailed understanding of event probability helps in solving problems like determining which CEOs earn more and how often shareholders achieve financial success.