Problem 52 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(E\) is an event of an experiment, then \(P(E)+P\left(E^{c}\right)=1\).

Step-by-Step Solution

Answer

The statement is true. The rule of complementary events states that the probabilities of an event and its complement must sum up to \(1\). Mathematically, this can be written as \(P(E) + P(E^c) = 1\), where \(E\) is an event of an experiment and \(E^c\) is its complement. Since \(E\) and \(E^c\) cover all the possible outcomes in the sample space, their probabilities must sum up to \(1\).

Step 1: Understanding Event and its Complement

Let \(E\) be an event of an experiment. Then, we can define its complement, denoted by \(E^c\), as the set of all outcomes in the sample space that are not in \(E\). In other words, when the experiment takes place, either \(E\) occurs or \(E^c\) occurs, but not both simultaneously.

Step 2: Rule of Complementary Events

The probabilities of complementary events sum up to \(1\). In mathematical terms, this can be written as: \[P(E) + P(E^c) = 1\]

Step 3: Explaining the Statement

The given statement states that if \(E\) is an event of an experiment, then \(P(E) + P(E^c) = 1\). This statement is true based on the rule of complementary events, which we stated earlier. Since \(E\) and \(E^c\) cover all the possible outcomes in the sample space (meaning one of them will always occur), their probabilities must sum up to \(1\).

Key Concepts

Complementary Events

Complementary events are a fundamental concept in probability theory, describing outcomes that collectively cover all possibilities in an experiment. Any given event, say \(E\), will have a complement, referred to as \(E^c\). This complement \(E^c\) consists of all the possible outcomes that are not part of the event \(E\) itself.

For example, if a dice is rolled, and \(E\) represents the event 'rolling an even number', then \(E^c\) would be 'rolling an odd number'. Such complementary events are mutually exclusive, meaning they cannot occur at the same time.

In any experiment, either one event or its complement will occur, but not both. This relationship highlights their complete coverage of the sample space and is captured mathematically as \(P(E) + P(E^c) = 1\).

Sample Space

Sample space refers to the set of all possible outcomes of a random experiment. Understanding the sample space is crucial for calculating probabilities accurately. Whenever an event occurs, it must be part of this larger collection of outcomes.

Think of the sample space like a bowl holding all the possible results of an experiment, whether it's flipping coins, rolling dice, or drawing cards. In practice, the sample space can be finite or infinite, and each potential outcome of our experiment is a member of this sample space.

• For a coin toss, the sample space would be \(\{\text{Heads}, \text{Tails}\}\).
• For a six-sided die, the sample space is \(\{1, 2, 3, 4, 5, 6\}\).

It's important to clearly define the sample space before analyzing any probabilities. Doing so ensures we have included all possible events and their respective complements.

Probability Formula

The probability formula is a key pillar in quantitative reasoning, enabling us to compute the likelihood of different events. When we talk about the probability of an event \(E\), denoted \(P(E)\), we are examining the fraction of the event's favorable outcomes relative to the total number of possible outcomes in the sample space.

The formula is given by: \[P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\]

When dealing with complementary events, the formula adapts to allow for simpler calculations. Using the complementary rule \(P(E) + P(E^c) = 1\), if you know the probability of one event, you can easily find its complement's probability with \(P(E^c) = 1 - P(E)\). This is particularly useful in cases where calculating \(P(E^c)\) is more straightforward than finding \(P(E)\) directly.

Thus, the probability formula serves as a versatile tool that equips us with necessary insights into possible experiment outcomes.