Problem 68 - Algebra for College Students

Question

If \(P(E)=\frac{5}{9}\) for some event \(E\), find the odds against \(E\) happening. 4 to 5

Step-by-Step Solution

Answer

The odds against event \(E\) happening are 4 to 5.

Step 1: Recall the Definition of Odds Against

To find the odds against an event \(E\) happening, we use the formula: **odds against** = \( \frac{1-P(E)}{P(E)} \). This formula converts the probability of \(E\) happening into the odds of \(E\) not happening relative to it happening.

Step 2: Calculate Probability of Event Not Happening

First, find the probability of \(E\) not happening, denoted as \(P(E')\). This is calculated using: \( P(E') = 1 - P(E) \). Substitute \(P(E) = \frac{5}{9}\), so \(P(E') = 1 - \frac{5}{9} = \frac{4}{9}\).

Step 3: Compute Odds Against Event E

Use the odds against formula: \( \frac{P(E')}{P(E)} = \frac{\frac{4}{9}}{\frac{5}{9}} \). Simplify this fraction to \( \frac{4}{5} \). The odds against \(E\) are 4 to 5.

Key Concepts

Understanding Probability

Probability is a fundamental concept in statistics and mathematics. It measures the likelihood of an event occurring. For any event \( E \), the probability \( P(E) \) is a value between 0 and 1.

• A probability of 0 means the event will not happen.
• A probability of 1 means the event is certain to happen.

A probability of **\( \frac{5}{9} \)**, as in our example, indicates that the likelihood of the event \( E \) occurring is relatively high, but not certain. This value provides a ratio that informs us about the balance between the chances of the event occurring versus not occurring.
It lays the groundwork for calculating other probabilities like odds against.

Defining an Event

In probability, an **event** is any result or outcome that may occur.

• An event can be something as simple as the flip of a coin resulting in heads.
• It could also be the occurrence of a highly specific outcome, like rolling two dice and getting a sum of ten.

Events can be single (simple events) or involve multiple possible outcomes (compound events). Understanding what constitutes an event is crucial as it sets the scope for determining its probability. In the given exercise, the event \( E \) has a probability of \( \frac{5}{9} \), meaning it is expected to occur more often than not when conditions are repeated."

Steps in Calculation

Calculation is the process of finding a numerical result using mathematical operations. When we are asked to find the **odds against** an event \( E \) occurring, we perform a sequence of calculations to derive the desired information. Here is how you perform these calculations:
1. **Find the complement** - Calculate the probability of the event not occurring \( P(E') \). This is done by subtracting \( P(E) \) from 1: \[ P(E') = 1 - P(E) \]For our event, this is: \[ P(E') = 1 - \frac{5}{9} = \frac{4}{9} \]
2. **Compute the odds against** - Use the probability of not occurring \( P(E') \) and divide it by \( P(E) \):\[ \text{odds against} = \frac{P(E')}{P(E)} = \frac{\frac{4}{9}}{\frac{5}{9}} = \frac{4}{5} \]These calculations give a clear numerical depiction of how likely the event is not to occur in comparison to it occurring.

Applying the Formula

Formulas are mathematical expressions that help us solve problems consistently and accurately. To find the odds against an event \( E\), the formula used is:\[ \text{odds against} = \frac{1-P(E)}{P(E)} \]This formula is crucial because it takes the probability of the event occurring, \( P(E) \), and transforms it into a ratio reflecting the likelihood of the event not happening.**Key Steps for Using the Formula**:

• Start with knowing \( P(E) \). In our context, it's \( \frac{5}{9} \).
• Find \( P(E') \) using \( 1 - P(E) \), which yields \( \frac{4}{9} \).
• Apply the formula by substituting the probabilities as shown above.

Following these steps ensures a proper understanding of how probabilistic formulas work, aiding in both comprehension and practical application in scenarios involving chances and outcomes.