Problem 24 - Discrete Mathematics with Applications

Question

Let \(U=|\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}|\) be the sample space of an experiment, where the outcomes are equally likely. Find the probability of each event. $$\\{\mathrm{a}, \mathrm{c}, \mathrm{d}\\}$$

Step-by-Step Solution

Answer

The probability of the event \(\{a, c, d\}\) is \(\frac{3}{5}\).

Step 1: Identify the event and sample space

The sample space, U, is given as |a, b, c, d, e|, and the event, A, is {a, c, d}.

Step 2: Count the number of outcomes in the sample space and event

The sample space U consists of five outcomes, so |U| = 5. The event A consists of three outcomes, so |A| = 3.

Step 3: Calculate the probability of the event

Since the outcomes in the sample space are equally likely, the probability of event A can be calculated by dividing the number of outcomes in A by the total number of outcomes in the sample space U: \[ P(A) = \frac{|A|}{|U|} \]

Step 4: Substitute the values and compute the probability

Now substitute the values of |A| and |U|: \[ P(A) = \frac{3}{5} \] So the probability of the event {a, c, d} is \(\frac{3}{5}\).

Key Concepts

Sample Space

The concept of a sample space is fundamental in the study of probability in discrete mathematics. It is the set of all possible outcomes of a random experiment. Each individual outcome within the sample space is unique and represents a different possible result of the experiment. For example, in the case of rolling a six-sided die, the sample space would be the set \( U = \{1, 2, 3, 4, 5, 6\} \), where each number corresponds to an outcome on the die.

Understanding the sample space is crucial because it serves as the foundation for calculating probabilities. It is important to identify all the elements of a sample space correctly to ensure that the probability calculations are accurate and meaningful. In the exercise provided, the given sample space \( U = \{a, b, c, d, e\} \) includes five distinct outcomes that can arise from the experiment being considered.

Equally Likely Outcomes

In probability, when we talk about equally likely outcomes, we refer to a scenario in which each outcome of an experiment has the same chance of occurring. This is a significant assumption because it simplifies the computation of probabilities, allowing us to use basic ratios and proportions. For applications like tossing a fair coin or rolling a fair die, this assumption holds because no single outcome has an advantage over the others.

When outcomes are equally likely, the probability of an event is calculated by dividing the number of outcomes in the event by the total number of outcomes in the sample space. This assumes a uniform distribution over the sample space, which is key to determining probabilities in many elementary scenarios in discrete mathematics. As given in the textbook exercise, the assumption that the outcomes \(a, b, c, d,\) and \(e\) are equally likely is central to finding the probability of such an event effectively and efficiently.

Event Probability Calculation

The event probability calculation is a method to determine the likelihood of a particular event occurring within the sample space. Once we have the sample space and understand the nature of the outcomes (whether they are equally likely or not), we can proceed to calculate the probability of an event. This is generally performed by subdividing the sample space into the desired event and its complement.

An event in probability is a set that contains zero or more outcomes of an experiment, and the probability of an event \(A\) is mathematically expressed as \( P(A) = \frac{\text{Number of outcomes in event A}}{\text{Total number of outcomes in sample space U}} \). To apply this formula, as illustrated in the exercise, count the number of outcomes in both the event and the sample space, then calculate their ratio. This is the foundational method for calculating probabilities, applicable across countless problems in discrete mathematics and beyond.