Problem 23 - 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. $$\\{a, b\\}$$

Step-by-Step Solution

Answer

The probability of the event {a, b} happening is \(\frac{2}{5}\) or 40%.

Step 1: Identify the size of the event

The given event is {a, b}. There are 2 elements (outcomes) in the event: a and b.

Step 2: Identify the size of the sample space

The sample space U has 5 equally likely outcomes: a, b, c, d, and e.

Step 3: Calculate the probability of the event

To find the probability of the event, divide the size of the event by the size of the sample space. The formula for probability can be written as: \(P(E) = \frac{\text{number of successful outcomes (size of event)}}{\text{total number of possible outcomes (size of sample space)}}\) Plug in the values we found in Steps 1 and 2: \(P(E) = \frac{2}{5}\)

Step 4: Simplify and present the final answer

The probability of the event {a, b} happening is \(\frac{2}{5}\) or 40%.

Key Concepts

Sample Space

In probability, a sample space refers to the set of all possible outcomes of a particular experiment. For the given problem, the sample space \(U\) consists of five outcomes: \(\{a, b, c, d, e\}\). These outcomes represent everything that could happen when the experiment is conducted. This foundational concept helps us organize and understand the range of potential results.

Understanding this space is crucial when calculating probabilities because it forms the denominator in all probability calculations. Knowing the entirety of what could happen lays the groundwork for determining what will happen in specific scenarios.

Equally Likely Outcomes

When outcomes are equally likely, each outcome in the sample space has the same chance of occurring. This is a basic assumption in many probability problems and simplifies the calculations. In the problem provided, outcomes \(a, b, c, d, e\) are described as equally likely. This means each has the same probability of being the result of the experiment.

To put it simply, the probability of any single outcome occurring is determined by dividing 1 by the total number of outcomes. For our sample space \(U\), each outcome has a probability of \(\frac{1}{5}\) because there are five equally possible outcomes. This assumption of equality makes it straightforward to calculate the probability of events involving these outcomes.

Event Probability

Event probability is the likelihood of a specific event occurring within the context of a sample space. It is calculated by dividing the number of successful outcomes by the total outcomes in the sample space. For instance, the event \(\{a, b\}\) in our problem has two successful outcomes: \(a\) and \(b\).

To calculate its probability, we use the formula:

• \(P(E) = \frac{\text{size of event}}{\text{size of sample space}} = \frac{2}{5}\)

This calculation reveals that there is a 40% chance of the event \(\{a, b\}\) occurring. By understanding event probability, you grasp how likely it is for certain outcomes to materialize.

Discrete Mathematics

Discrete mathematics plays a vital role in understanding probability concepts. It deals with countable, distinct elements, much like the outcomes in our sample space \(U\). The principles of discrete mathematics help us manage and calculate probabilities where simple, distinct outcomes are considered.

In this context, probability problems are often grounded in the clear, defined operations that discrete mathematics supports. By analyzing situations through discrete structures, such as sets and events, we can clearly ascertain probabilities and make logical deductions. This systematic approach is integral to solving probability problems with fair and precise methods.