Welcome back to class! In this lesson, Section 8-5, we are exploring the fascinating world of Probability. While we often hear these terms used interchangeably in daily life, in mathematics, there is a distinct and important difference between "probability" and "odds." Today, we will learn how to calculate both and convert between them.

1. The Basic Definition of Probability

Probability is the measure of how likely an event is to occur. It is always a ratio comparing the number of specific successful outcomes to the total number of possible outcomes. The probability of an event $E$ is expressed as:

$$P(E) = \frac{\text{Number of Successes}}{\text{Total Number of Possible Outcomes}}$$

Probabilities are usually written as fractions or decimals between $0$ and $1$, or as percentages between $0\%$ and $100\%$.

2. Understanding Odds

Unlike probability, which compares a part to the whole, Odds compare a part to the remaining part. We look at the ratio of successes to failures.

  • Odds in Favor: The ratio of favorable outcomes to unfavorable outcomes. $$ \text{Odds}(E) = \frac{\text{Successes}}{\text{Failures}} $$
  • Odds Against: The ratio of unfavorable outcomes to favorable outcomes. $$ \text{Odds Against}(E) = \frac{\text{Failures}}{\text{Successes}} $$

3. Converting Between Probability and Odds

A key skill in this section is converting a probability percentage into an odds ratio. Let's look at a classic example involving weather.

Example: If the weatherman says there is a $20\%$ probability of rain, what are the odds in favor of rain?

  1. First, convert the percentage to a fraction: $20\% = \frac{20}{100} = \frac{1}{5}$.
  2. Identify the parts: The Total is $5$. The Success (rain) is $1$.
  3. Calculate the Failures: $\text{Total} - \text{Success} = \text{Failures}$. So, $5 - 1 = 4$.
  4. Write the Odds: The odds in favor are $1:4$ (read as "1 to 4").

4. Study Tips

As you work through the attached workbook and review the video transcript, pay close attention to the wording of the questions. The most common mistake is confusing the denominator—remember that probability uses the Total, while odds use the Failures.

Good luck with the practice problems! Mastering these conversions now will make future sections on complex events much easier.