Question

Consider a sample space \(\Omega=\\{a, e, i, o, u\\}\) endowed with the following probability density: \(p(a)=0.22, p(e)=0.35, p(i)=0.13, p(o)=0.20,\) and \(p(u)=0.10 .\) Determine the probabilities of the following events: (a) \(\\{a, o\\}\) (b) \(\emptyset\) (c) The event \(E\) consisting of all those outcomes in \(\Omega\) that come after the letter \(k\) in the alphabet

Short answer

(a) 0.42; (b) 0; (c) 0.30.

Step-by-step solution

Understand the Given Data

We are given a sample space \(\Omega = \{a, e, i, o, u\}\) with the probabilities for each element: \(p(a) = 0.22\), \(p(e) = 0.35\), \(p(i) = 0.13\), \(p(o) = 0.20\), and \(p(u) = 0.10\). Our task is to find the probability of different events described using this sample space.

Calculate Probability of Event \(\{a, o\}\)

Event \(\{a, o\}\) consists of the outcomes \(a\) and \(o\). The probability of an event is the sum of the probabilities of the individual outcomes in the event. Thus: \[P(\{a, o\}) = p(a) + p(o) = 0.22 + 0.20 = 0.42.\]

Calculate Probability of the Empty Set \(\emptyset\)

The empty set \(\emptyset\) contains no outcomes. By definition, the probability of the empty set is always 0: \[P(\emptyset) = 0.\]

Define and Calculate Probability for Event \(E\)

Event \(E\) includes all outcomes in \(\Omega\) that come after the letter \(k\). In the alphabet, the letters \(o\) and \(u\) come after \(k\), so \(E = \{o, u\}\). Calculate the probability: \[P(E) = p(o) + p(u) = 0.20 + 0.10 = 0.30.\]

Key concepts

Probability Density

Probability density is an essential concept in understanding the likelihood of outcomes in a sample space. In this context, a sample space is a set of all possible outcomes of an experiment or situation. Here, our sample space is \(\Omega = \{a, e, i, o, u\}\), each with its own probability.
Probability density is a function that assigns a probability to each outcome in the sample space. In the given example, probabilities are provided for each element:

• \(p(a) = 0.22\)
• \(p(e) = 0.35\)
• \(p(i) = 0.13\)
• \(p(o) = 0.20\)
• \(p(u) = 0.10\)

The sum of these probabilities must equal 1, ensuring that they map out the entire sample space. Understanding how to assign and manipulate these probabilities is crucial in calculating the probability of various events.

Events

In probability theory, an event refers to a subset of outcomes within the sample space. Events can be simple or compound, and their probability is calculated by summing up the probabilities of their respective outcomes.
For instance, we determined the probability of event \(\{a, o\}\), which includes the outcomes 'a' and 'o'. Thus, the probability of the event is computed as:\[P(\{a, o\}) = p(a) + p(o) = 0.22 + 0.20 = 0.42.\]Events can vary broadly and can include specific features, such as containing particular letters or conditions.
In solving problems, clearly defining the events helps simplify the calculation by breaking down the sample space into manageable parts.

Alphabetical Order

Alphabetical order often aids in determining conditions within a sample space. It is a straightforward method where items are arranged in alphabetical sequence, which can be helpful for understanding positions or sequences of characters or events.
For our exercise, part of determining probabilities involved distinguishing outcomes that come after a particular letter, \'k\', alphabetically within the set \(\Omega = \{a, e, i, o, u\}\).
The event described as outcomes coming after \'k\' resulted in \'o\' and \'u\'. Understanding this sequence allowed us to form event \(E\), where \(E = \{o, u\}\), making it easy to add up the respective probabilities to find:\[P(E) = p(o) + p(u) = 0.20 + 0.10 = 0.30.\]Alphabetical order is not just for arranging characters but becomes a useful tool in sorting and evaluating conditions on data sets.

Empty Set

The empty set, denoted as \(\emptyset\), is a fundamental notion in set theory. It contains no elements and is a unique set with a probability of zero in probability theory.
Calculating the probability of the empty set is always straightforward, as it does not represent any outcomes or scenarios within the sample space.
By definition,\[P(\emptyset) = 0.\]This underscores that without any possible events occurring, there's no likelihood of such a scenario unfolding.
In practice, the concept of the empty set is useful for representing scenarios or conditions where no outcomes meet specific criteria, ensuring clarity and precision in probability calculations.