91Ó°ÊÓ

Betting odds are usually stated against the event happening (against winning). The odds against event \(W\) is the ratio \(\frac{P(n o t W)}{P(W)}=\frac{P\left(W^{c}\right)}{P(W)}\). In horse racing, the betting odds are based on the probability that the horse does not win. (a) Show that if we are given the odds against an event \(W\) as \(a: b\), the probability of not \(W\) is \(P\left(W^{c}\right)=\frac{a}{a+b} \cdot\) Hint \(:\) Solve the equation \(\frac{a}{b}=\frac{P\left(W^{c}\right)}{1-P\left(W^{c}\right)}\) for \(P\left(W^{c}\right)\). (b) In a recent Kentucky Derby, the betting odds for the favorite horse, Point Given, were 9 to \(5 .\) Use these odds to compute the probability that Point Given would lose the race. What is the probability that Point Given would win the race? (c) In the same race, the betting odds for the horse Monarchos were 6 to 1 . Use these odds to estimate the probability that Monarchos would lose the race. What is the probability that Monarchos would win the race? (d) Invisible Ink was a long shot, with betting odds of 30 to 1 . Use these odds to estimate the probability that Invisible Ink would lose the race. What is the probability the horse would win the race? For further information on the Kentucky Derby, visit the Brase/Brase statistics site at college .hmco.com/pic/braseUs9e and find the link to the Kentucky Derby.

Step by step solution

01

Understand the relationship between odds and probabilities

The odds against an event \(W\) are given as \(a: b\). This means the probability of the event not happening \(P(W^c)\) is compared to the probability of the event happening \(P(W)\) as \(\frac{a}{b}\). This can be rewritten using probabilities as \(\frac{P(W^c)}{1 - P(W^c)}\) because \(P(W^c) + P(W) = 1\).

02

Solve for Probability of event not happening

To find \(P(W^c)\), solve the equation \(\frac{a}{b} = \frac{P(W^c)}{1 - P(W^c)}\). Cross-multiply to get \(a(1 - P(W^c)) = bP(W^c)\). Simplify to \(a - aP(W^c) = bP(W^c)\). Rearrange to \(a = aP(W^c) + bP(W^c) = P(W^c)(a + b)\). Thus, \(P(W^c) = \frac{a}{a + b}\).

03

Calculate Probability for Point Given

Given betting odds of 9 to 5 for Point Given, use the formula \(P(W^c) = \frac{a}{a + b}\). Here, \(a = 9\) and \(b = 5\). Therefore, \(P(W^c) = \frac{9}{9 + 5} = \frac{9}{14}\). \(P(W) = 1 - P(W^c) = 1 - \frac{9}{14} = \frac{5}{14}\).

04

Calculate Probability for Monarchos

Given betting odds of 6 to 1 for Monarchos, use the formula \(P(W^c) = \frac{6}{6 + 1} = \frac{6}{7}\). Therefore, \(P(W) = 1 - \frac{6}{7} = \frac{1}{7}\).

05

Calculate Probability for Invisible Ink

Given betting odds of 30 to 1 for Invisible Ink, use the formula \(P(W^c) = \frac{30}{30 + 1} = \frac{30}{31}\). Therefore, \(P(W) = 1 - \frac{30}{31} = \frac{1}{31}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horse Racing Odds

In horse racing, the term "odds" holds a special significance. These odds are presented in the format such as 9:5, 6:1, or 30:1, and they reflect the bookmaker's assessment of a horse's chances to not win a race, which can be initially confusing. Here's how it works:

Within this ratio, the first number tells you how much you can win, while the second number indicates your stake. If the odds against a horse are 9 to 5, this means you might win 9 units for every 5 units you bet, but only if the horse does not win. If the horse does win, those odds are great for the bookmaker, not for the bettor.

• The odds reflect the probability of the horse not winning. Hence, high odds often indicate a low chance of the horse winning the race.
• Understanding this terminology is crucial for bettors as it impacts their decision-making process during wagers.

In essence, betting odds in horse racing present a fascinating way to predict race outcomes as they consider both the horse's past performance and public sentiment.

Betting Odds

When discussing betting odds in general, they are more than just numbers. They represent the likelihood of an event happening in comparison to another outcome. Odds are expressed in different formats such as fractional, decimal, or American odds, but they all convey the same underlying idea.

In scenarios like sports betting:

• Odds show the bookmaker's confidence in the outcome and guide bettors on potential winnings.
• Fractions such as 9/5 (or 9:5) can be converted to probability through the formula:
  \( P\left(W^{c}\right) = \frac{a}{a+b} \)
  
• This means you assess not only the potential rewards but also the likelihood of the event happening.

The clearer grasp you have of these figures, the more informed decisions you can make in your betting strategy.

Odds Against an Event

Odds against an event happening essentially provide a perspective on the probability of something not occurring. For instance, calculating the odds against a horse winning a race involves comparing the probability of the horse losing to it winning. This can be elucidated through the equation:

\[\frac{P\left(W^{c}\right)}{1-P\left(W^{c}\right)} = \frac{a}{b} \]
This provides insights into chances against the event happening versus it happening. When dealing with odds, here is what each part represents:

• \(P(W^{c})\): The probability of the event not happening.
• \(P(W)\): The probability of the event happening.
• Rearranging and calculating using \(P(W^{c}) = \frac{a}{a+b}\) helps in expressing these odds in a more intuitive way.

This formula is key to evaluating betting odds in both daily life scenarios and complex betting systems, cementing its importance in probability theory.