Problem 72 - Algebra for College Students

Question

The odds against Belly Dancer winning the fifth race are 20 to 9 . What is the probability of Belly Dancer winning the fifth race? \(\quad P=\frac{9}{29}\)

Step-by-Step Solution

Answer

The probability of Belly Dancer winning is \( \frac{9}{29} \).

Step 1: Understand Odds Against

Odds against an event occur when the probability of the event happening is expressed as a ratio of unsuccessful to successful outcomes. Here, the odds against Belly Dancer winning are 20 to 9, meaning there are 20 chances of losing for every 9 chances of winning.

Step 2: Convert Odds to Probability

Probability is the ratio of the successful outcomes to the total number of outcomes. The total number of outcomes is the sum of the lose and win outcomes, given by 20 (lose) + 9 (win) = 29 outcomes.

Step 3: Calculate Probability of Winning

The probability of winning is the successful outcomes (wins) divided by the total outcomes: \( P(\text{win}) = \frac{9}{29} \). This value represents the probability of Belly Dancer winning the race based on the given odds.

Key Concepts

Understanding Odds Against

When we talk about the "odds against" an event occurring, we're looking at a comparison of the likelihood of the event not happening versus it happening. In other words, it's a way to express how likely failure is compared to success. For instance, if the odds against Belly Dancer winning a race are 20 to 9, this means for every 9 chances Belly Dancer has to win, there are 20 chances she will not win. This ratio is crucial to understanding what's more likely: the event or its failure.

Odds against are important because

• they help quantify risk in probabilistic scenarios,
• they help us understand the implied probability of certain outcomes,
• they serve as a basis for calculating probabilities more formally.

Understanding odds provides a solid foundation when we're moving into the world of probability calculations, as odds describe the balance between winning and losing possibilities.

Calculating Probability of Winning

Once we have the odds against an event, we can convert these into a probability of winning. This involves a straightforward formula which helps us determine how likely a successful outcome is. The probability (\(P\)) of winning is calculated by dividing the number of successful outcomes by the total number of possible outcomes.

In the given example, Belly Dancer's odds of winning are 9 successful outcomes against 29 total outcomes, which includes 20 losses and 9 wins. So, the probability of winning (\(P\)) is given by \(P = \frac{9}{29}\). This result, \(P\), tells us that if the race were run repeatedly, Belly Dancer would win approximately \(31 ext{%}\) of the time. It’s important to notice how a strong grasp on both the odds and the potential number of outcomes aids in understanding probability effectively.

Understanding the Ratio of Outcomes

The ratio of outcomes is foundational to understanding both odds and probability. This concept refers to comparing the different ways something can happen, be it success or failure. It provides a snapshot of all possible outcomes and helps in structuring our understanding of chance.

For example, with odds given as 20 to 9 against Belly Dancer, there are two groups of outcomes:

• 20 chances of losing,
• 9 chances of winning.

This ratio, 20:9, becomes 20 divided by 9 when calculating odds against. However, total outcomes matter as well. By adding both success and failure outcomes, we get the total number of opportunities, which forms the basis for calculating probability. For instance, with 29 total potential results (20 loses + 9 wins), we use this number to determine how likely a single win is.

By understanding the ratio of different outcomes, we not only grasp the mechanics of probability but also gain clearer insights into what those odds are telling us in any given scenario.