Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

If \(P(E)=0.4\), find \(O(E)\) and \(O\left(E^{\prime}\right)\)

Step-by-Step Solution

Answer

\(O(E) = \frac{2}{3}\); \(O(E') = \frac{3}{2}\).

Step 1: Understanding Probability and Odds

First, let's remember the relationship between probability and odds. The probability of an event, denoted as \(P(E)\), is the chance that an event will occur. On the other hand, odds (\(O(E)\)) represent the ratio of the probability that the event will occur to the probability that it will not occur. If \(P(E) = 0.4\), the formula to find odds is \(O(E) = \frac{P(E)}{1-P(E)}\).

Step 2: Calculate Odds for Event E

We know that \(P(E) = 0.4\). Using the formula for odds:\[O(E) = \frac{0.4}{1-0.4}\]Calculate as follows:\[O(E) = \frac{0.4}{0.6} = \frac{2}{3}\]So, the odds for event \(E\) is \(\frac{2}{3}\).

Step 3: Calculate the Complement Probability

The complement of event \(E\), denoted \(E'\), is the event that \(E\) does not occur. The probability of the complement is calculated as:\[P(E') = 1 - P(E)\]Given \(P(E) = 0.4\), we have:\[P(E') = 1 - 0.4 = 0.6\].

Step 4: Calculate Odds for the Complement Event

Using the same odds calculation method for the complement event \(E'\):\[O(E') = \frac{P(E')}{1-P(E')}\]Substitute the value of \(P(E')\):\[O(E') = \frac{0.6}{1-0.6} = \frac{0.6}{0.4} = \frac{3}{2}\]So, the odds for \(E'\) is \(\frac{3}{2}\).

Key Concepts

Odds Calculation

Understanding how to calculate odds can help in many real-world situations, from betting to statistical analysis. Odds provide a different perspective than just looking at probabilities.

• Definition: Odds (O(E)) represent the likelihood ratio of an event occurring compared to it not occurring.
• Formula: To calculate the odds of an event, use the formula \(O(E) = \frac{P(E)}{1-P(E)}\), where \(P(E)\) is the probability of the event.
• Example: For a probability of an event \(P(E) = 0.4\), the odds are calculated as \(O(E) = \frac{0.4}{0.6} = \frac{2}{3}\).

Odds can often be more intuitive when making decisions, as they speak to the balance between occurrence and non-occurrence. This makes them particularly useful in situations where weighing different chances is required.

Complement Probability

The concept of complement probability encourages considering both the event and its non-occurrence. This view can be both practical and necessary.

• Definition: The complement of an event \(E\), denoted as \(E'\), is the scenario where event \(E\) does not happen.
• Relation: The sum of an event's probability and its complement always equals 1: \(P(E) + P(E') = 1\).
• Example: Given \(P(E) = 0.4\), the complement probability is \(P(E') = 1 - 0.4 = 0.6\).

When planning for uncertain events, understanding complement probabilities can ensure a more complete assessment of potential outcomes, effectively providing the full picture.

Probability of an Event

Probability is a fundamental concept in understanding chance and risk in everyday situations. It provides a foundation for statistical reasoning.

• Definition: The probability of an event \(E\), denoted as \(P(E)\), measures how likely the event is to occur, ranging from 0 (impossible) to 1 (certain).
• Calculation: In practice, probability is calculated as the number of successful outcomes divided by the total number of possible outcomes.
• Example: If \(P(E) = 0.4\), this indicates a 40% chance that event \(E\) will happen.

This concept helps in decision-making processes by quantifying levels of risk and enabling comparison amongst possible events, making better informed predictions.