Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Explain why the statement is incorrect. The probability that a certain stock will increase in value over a period of 1 week is .6. Therefore, the probability that the stock will decrease in value is \(.4\)

Step-by-Step Solution

Answer

The given statement is incorrect because it assumes that the stock value increasing and decreasing are mutually exclusive and complementary events, neglecting the possibility of the stock value remaining the same. A more accurate statement would consider the probability of the stock value remaining the same (X), with the probability of the stock decreasing in value being \(1 - 0.6 - X\).

Step 1: Understand complementary events:

In probability theory, complementary events are events that cannot both occur simultaneously. An event and its complementary event sum up to a total probability of 1. In other words, the probability that either event A or its complement (not event A) occurs is always 1. Mathematically, this can be represented as \[ P(A) + P(\overline{A}) = 1, \] where \(P(\overline{A})\) is the probability of the complementary event of event A not occurring.

Step 2: Determine mutual exclusivity:

The next step is to determine whether the events of the stock value increasing and decreasing are mutually exclusive or not. Mutually exclusive events are events that cannot both occur at the same time, meaning if one happens, the other cannot happen at the same time. If the events are mutually exclusive, then the probability of the stock value decreasing can be considered as the complementary event, and the given statement would be correct.

Step 3: Analyze events in relation to the stock value:

There are three possibilities in the context of the stock value: 1. The stock value increases. 2. The stock value decreases. 3. The stock value remains the same. If the stock value either increases or decreases, then it cannot remain constant. However, if the statement assumes the probability of the stock value decreasing is the complement to the stock value increasing (ignoring the possibility of the stock value staying the same), then the statement is incorrect.

Step 4: Conclusion:

The given statement is incorrect because it assumes that the stock value increasing and decreasing are mutually exclusive and complementary events, which is not true considering a third possibility of the stock value remaining the same. The probability of the stock decreasing in value could be less than, equal to, or greater than \(0.4\). A more accurate statement would be: if the probability that a certain stock will increase in value over a period of 1 week is \(0.6\) and the probability that the stock value remains the same is \(X\), then the probability that the stock will decrease in value is \(1 - 0.6 - X\).

Key Concepts

Complementary Events

Understanding complementary events is essential in the realm of probability theory. These events represent a fundamental relationship: one cannot happen if the other has occurred. To put it simply, if we have an event A, the complementary event, often denoted as \( \overline{A} \) or 'not A', encompasses all outcomes that are not part of A. For any event, the sum of the probabilities of the event and its complementary event equals one, which is expressed mathematically as \[ P(A) + P(\overline{A}) = 1. \]

In practical terms, if we know the probability of it raining today is 0.3, then the probability of it not raining (\

Mutually Exclusive Events

As we dive into the topic of mutually exclusive events, it's important to grasp that these are different from complementary events. While mutually exclusive events also cannot occur at the same time, they do not necessarily cover all possible outcomes when combined. In other words, two events are considered mutually exclusive if the happening of one event rules out the occurrence of the other. Keeping this in mind, let's look at a simple example: flipping a coin. The results of 'heads' and 'tails' are mutually exclusive because getting a 'head' means you cannot simultaneously get a 'tail', and vice versa.

However, mutual exclusivity doesn't imply that the probabilities of the events must add up to one. For instance, if you roll a die, the events of rolling a '1' and rolling a '2' are mutually exclusive, but their combined probabilities do not account for all possible outcomes of the die roll. Recognizing these distinctions can prevent misconceptions during probability calculation and ensures accuracy in reasoning about random events.

Probability Calculation

Getting into the nitty-gritty of probability calculation, this process is often the final step in understanding random events. It involves working with known probabilities to deduce unknown probabilities or to assess the likelihood of a series of events. The exercise posed about the stock price illustrates a common error in probability calculation: assuming there are only two outcomes without considering other possibilities. Here, the error was in neglecting the chance of the stock value staying the same.

Correcting such oversights involves identifying all possible outcomes and ensuring that their probabilities account for the full range of events. In practice, when the known probability that a stock will increase in value is 0.6, we cannot immediately conclude that the probability of a decrease is 0.4 without knowing the probability of the stock value remaining unchanged. Think of it as a complete picture; each fragment (event) must be accounted for to understand the whole (total probability). Incorporating principles from complementary and mutually exclusive events, as well as considering all possibilities, underpins accurate probability calculation.