Problem 14 - Basic Statistics for Business and Economics with Student

Question

The chair of the board of directors says, "There is a 50 percent chance this company will earn a profit, a 30 percent chance it will break even, and a 20 percent chance it will lose money next quarter." a. Use an addition rule to find the probability the company will not lose money next quarter. b. Use the complement rule to find the probability it will not lose money next quarter.

Step-by-Step Solution

Answer

Probability the company will not lose money is 0.80.

Step 1: Understanding the Problem

We are given the probabilities of three mutually exclusive events related to the company's financial outcome: earning a profit (50%), breaking even (30%), and losing money (20%). We need to find the probability of the company not losing money, which means either earning a profit or breaking even.

Step 2: Using the Addition Rule

To find the probability of the company not losing money using the addition rule, add the probabilities of the mutually exclusive events of earning a profit and breaking even. Calculate it as: \( P(\text{Profit}) + P(\text{Break Even}) = 0.50 + 0.30 = 0.80 \). The probability the company will not lose money is 0.80.

Step 3: Using the Complement Rule

The complement rule states that the probability of an event not occurring is 1 minus the probability of the event occurring. For the company losing money, the probability is 0.20. Therefore, the probability the company will not lose money is \( 1 - P(\text{Lose Money}) = 1 - 0.20 = 0.80 \).

Key Concepts

Addition Rule

The addition rule in probability theory is particularly useful when we want to determine the probability of the occurrence of at least one of several events. In contexts like the provided exercise, it helps in calculating the likelihood that a particular outcome takes place, without having to analyze each outcome separately.

There are two categories within the addition rule:

• If the events are mutually exclusive, the addition rule is straightforward. The probability of either event A or event B occurring is the sum of the probabilities of each event: \( P(A \cup B) = P(A) + P(B) \).
• If the events are not mutually exclusive, you must subtract the probability of both events occurring from their sum, i.e., \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).

In the exercise, earning a profit and breaking even are mutually exclusive, meaning they cannot both occur simultaneously. Thus, the probability of not losing money, which is the union of these events, is simply the sum of their probabilities.

Remember, using the addition rule is intuitive when you ensure no event overlap, as in mutually exclusive settings, and it ensures the total occurrences are considered accurately.

Complement Rule

Understanding the complement rule is essential in probability theory, as it simplifies solving certain types of problems immensely. The rule essentially states that the probability of an event not occurring equals one minus the probability of the event occurring.

This can be expressed in formula terms as: \( P( ext{Not A}) = 1 - P(A) \). This is particularly useful when you know the probability of a specific event and want the probability of the opposite. It's a quick check to ensure all eventualities are considered.

In the context of the exercise, we used the complement rule to find out the probability that the company will not lose money. Since we know the probability that it will in fact lose money is 0.20, the probability it won’t is \( 1 - 0.20 = 0.80 \).

This method provides an alternative check and often easier calculation for events where the probability of not requiring detailed enumeration. So, always remember, sometimes asking 'what's the opposite chance?' is the simplest route to a solution.

Mutually Exclusive Events

In probability, when we refer to mutually exclusive events, we're talking about events that cannot occur at the same time. For example, flipping a coin results in either heads or tails, never both.

Mutually exclusive events are fundamental in understanding probability calculations because they simplify the use of the addition rule. Suppose events A and B are mutually exclusive, then the occurrence of A excludes B, and vice-versa. It means \( P(A \cap B) = 0 \).

Using mutually exclusive principles helps in blending different outcomes into a single calculation without overlap. In the provided exercise, the company earning a profit and breaking even are mutually exclusive scenarios. Therefore, you can easily calculate the probability of not losing money by simply adding the probabilities of earning a profit and breaking even.

Recognizing mutually exclusive events helps prevent miscalculations due to double counting in more complex scenarios. It's essential when setting up probability models that correctly reflect real-world logic and restrictions.