Problem 51 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(E\) and \(F\) be events such that \(F \subset E\). Find \(P(E \mid F)\) and interpret your result.

Step-by-Step Solution

Answer

The conditional probability \(P(E \mid F)\) is 1. This means if event F occurs, then event E is certain to occur, since F is a subset of E and all outcomes in F are included in E.

Step 1: Identify the intersection of events E and F

Since F is a subset of E, it means that all outcomes in event F are also included in event E. Therefore, the intersection of E and F, denoted by E ∩ F, is simply F itself: \[E \cap F = F\]

Step 2: Calculate the probabilities P(E ∩ F) and P(F)

Given that \(E \cap F = F\), for the numerator, the probability of the intersection is the same as the probability of F: \[P(E \cap F) = P(F)\] As for the denominator, we don't have any information about P(F) from the given exercise. So, we'll leave it as is, denoting it as P(F).

Step 3: Apply the formula for conditional probability

Now that we have the probabilities for \(P(E \cap F)\) and \(P(F)\), we can apply the formula for conditional probability: \[P(E \mid F) = \frac{P(E \cap F)}{P(F)}\] By substituting \(P(E \cap F) = P(F)\): \[P(E \mid F) = \frac{P(F)}{P(F)}\] Since P(F) > 0, we can simplify this expression: \[P(E \mid F) = 1\]

Step 4: Interpret the result

The result P(E | F) = 1 indicates that if event F occurs, then event E is certain to occur as well. This is consistent with the given condition that F is a subset of E, meaning once we know F has occurred, E must also have occurred since all outcomes in F are included in E.

Key Concepts

Probability Theory

Probability theory is the branch of mathematics that deals with the analysis of random events. The main goal is to determine the likelihood of various outcomes.

At the core of probability is the idea that we can quantify how likely it is for an event to occur. This is given as a number between 0 and 1, where 0 means the event will not occur, and 1 indicates certainty that the event will occur. Probabilities can be based on observations, experimental data, or theoretical principles.

In this case, understanding probability also involves understanding what happens when we have information about related events. Conditional probability, which appears in our exercise, is a measure of the probability of an event occurring given that another event has already occurred.

Subset of Events

In probability theory, when we talk about a 'subset of events,' we are referring to how one event, let's call it Event F, can be entirely contained within another event, known as Event E. This is denoted as F \(\subset E\), meaning every outcome in F is also an outcome in E.

If Event F happens, Event E must also happen since Event F's outcomes are part of Event E's outcomes. For instance, if Event E is 'drawing a face card in a deck of playing cards' and Event F is 'drawing the Queen of Hearts', then F is a subset of E as the Queen of Hearts is one of the face cards.

Intersection of Events

The intersection of events, denoted as E \(\cap F\), is the set of outcomes that two events have in common. This represents the scenario where both events occur simultaneously.

In our textbook problem, since Event F is a subset of Event E, the intersection of E and F is just F. We write E \(\cap F = F\). The probability of their intersection P(E \(\cap F\)) is equivalent to the probability of F because all the outcomes of Event F must occur if Event E occurs.

Understanding the intersection of events is crucial when calculating conditional probabilities since it represents the outcomes that are relevant to both conditions being true.