Problem 22 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The probability of event \(\{a\}\) is \(\frac{1}{5}\).

Step 1: Determine the number of outcomes in the sample space and event

We first need to establish the number of outcomes in the sample space (\(U\)) and the event (\(\{a\}\)). The number of outcomes in the sample space is \(|U| = 5\) (since there are 5 outcomes: \(a\), \(b\), \(c\), \(d\), and \(e\)). The event \(\{a\}\) has only one outcome, so its number of outcomes is \(|\{a\}| = 1\).

Step 2: Probability formula for equally likely outcomes

In situations with equally likely outcomes, we can use the following formula to find the probability of an event \(E\): $$P(E) = \frac{|E|}{|U|}$$ In this case, the event \(E\) is \(\{a\}\) so we can write: $$P(\{a\}) = \frac{|\{a\}|}{|U|}$$

Step 3: Compute the probability

Now, we will use the formula to calculate the probability of event \(\{a\}\): $$P(\{a\}) = \frac{|\{a\}|}{|U|} = \frac{1}{5}$$ So, the probability of event \(\{a\}\) is \(\frac{1}{5}\).

Key Concepts

Sample Space

A sample space is a fundamental concept in probability that encompasses all possible outcomes of an experiment. When we talk about a sample space, we are simply referring to a set that includes every potential result that could occur from a particular experiment or random trial.
In our original exercise, the sample space is denoted as \(U = \{a, b, c, d, e\}\), which contains all outcomes this specific experiment can yield.
Understanding the sample space is critical as it forms the basis for determining probabilities. By observing the sample space, you can identify and count the total number of possible outcomes, which is crucial for probability calculations. In this case, there are 5 possible outcomes in the sample space.

Equally Likely Outcomes

Equally likely outcomes refer to a scenario where each possible result of an experiment has the same chance of occurring. This is a common assumption in theoretical probability, simplifying calculations.
In the given problem, we are told all outcomes \(a, b, c, d, e\) are equally likely. This means each has the same likelihood of getting chosen, making it easier to apply probability formulas.

• For example, if you roll a fair die, each number from 1 to 6 has an equal 1/6 chance of being rolled, exemplifying equally likely outcomes.
• In our sample space \(U\), the equal likelihood implies each event consisting of one outcome has the probability that can be directly calculated by counting.

This setup is essential for using standard probability formulas effectively.

Probability Formula

The probability formula for equally likely outcomes is a straightforward way to calculate the chance of a specific event happening, based on the assumption that all outcomes are equally probable.
The formula is expressed as: \[P(E) = \frac{|E|}{|U|}\]
Here, \(|E|\) is the number of favorable outcomes for the event \(E\), and \(|U|\) is the total number of outcomes in the sample space. This formula reflects the ratio of favorable outcomes to the total, providing a clear probability value.
In the problem at hand, this formula helps calculate the probability of the event \(\{a\}\): since \(|\{a\}| = 1\) and \(|U| = 5\), we compute \[P(\{a\}) = \frac{1}{5}\] This means there is a 1 in 5 chance, or 20%, of the event occurring, given each outcome is equally likely.