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Chapter 1: Problem 13

In a horse race, the odds that Romance will win are listed as 2: 3 and that Downhill will win are 1: 2 . What odds should be given for the event that either Romance or Downhill wins?

Short Answer

Expert verified
3:2

Step by step solution

01

Understanding Odds

Odds of 2:3 for Romance winning means for every 2 times Romance wins, it loses 3 times. Thus, the probability of Romance winning is \( \frac{2}{2+3} = \frac{2}{5} \). Similarly, odds of 1:2 for Downhill winning means the probability of Downhill winning is \( \frac{1}{1+2} = \frac{1}{3} \).
02

Calculating Total Probability

To find the probability that either Romance or Downhill wins, consider them as two independent events. The probability that either wins is given by \( P(R ext{ or } D) = P(R) + P(D) - P(R)P(D) \). Substitute the probabilities: \( P(R ext{ or } D) = \frac{2}{5} + \frac{1}{3} - \left(\frac{2}{5} \times \frac{1}{3}\right) \).
03

Simplifying the Equation

First, find a common denominator for \( \frac{2}{5} \) and \( \frac{1}{3} \), which is 15. Rewrite: \( \frac{6}{15} + \frac{5}{15} - \left(\frac{2}{15}\right) \).
04

Solving the Combined Probability

Combine the fractions: \( \frac{6}{15} + \frac{5}{15} = \frac{11}{15} \). Now, subtract the intersection: \( \frac{11}{15} - \frac{2}{15} = \frac{9}{15} \), which simplifies to \( \frac{3}{5} \).
05

Converting Probability to Odds

We have \( \frac{3}{5} \) as the total probability for either Romance or Downhill winning. The odds in favor are \( \text{wins : total events - wins} \) which equals \( 3 : (5 - 3) = 3:2 \).

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Key Concepts

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

Understanding Odds
Odds are a way of expressing the likelihood of an event occurring. They are typically presented in the form "x:y", where "x" represents the number of successful outcomes, and "y" represents the number of unsuccessful outcomes. For instance, if the odds for a horse named Romance winning a race are 2:3, it means for every 2 times Romance wins, it doesn't win 3 times. This relationship helps us convert odds into a more standard probability format.
  • Calculate the probability by summing the odds: add the two parts of the odds ratio together. For 2:3, this sum is 5.
  • Then, determine the probability of the event by dividing the winning part by this total: hence, the probability Romance wins is \( \frac{2}{5} \).
Understanding this conversion is essential when combining probabilities from multiple, independent events like in a horse race.
Probability Calculation
Calculating probabilities involves breaking down complex problems into more straightforward steps. When you have multiple independent events, like Romance or Downhill winning, you need to calculate their combined probability.
To find the probability that either Romance or Downhill wins, treat them as independent. The formula used is:\[ P(R \text{ or } D) = P(R) + P(D) - P(R)P(D) \]This equation accounts for both events occurring independently.
  • First, calculate each individual probability from their given odds.
  • Then, use the addition rule for independent events which subtracts the probability of both occurring together, as it gets double-counted.
  • Substitute the probabilities known from odds: Romance鈥檚 \( \frac{2}{5} \) and Downhill鈥檚 \( \frac{1}{3} \).
Calculate the fractions with a common denominator to ensure the math is straightforward and accurate. Combine and simplify the fractions to get the final probability that either Romance or Downhill will win.
Independent Events
When handling independent events, it's crucial to understand how their probabilities interact. Independent events are scenarios where the outcome of one event does not affect the other. In our horse race example, Romance winning does not change the probability of Downhill winning.
  • Recognize the independence by identifying there is no overlap in their outcomes.
  • Use the probability formula for independent events to find the combined probability.
  • Subtract the chance both occur simultaneously 鈥� calculated as the product of their probabilities 鈥� to prevent overestimating the odds.

      • By appreciating the role of independent events, you ensure precise probability calculations, reflecting the true chance of at least one occurring. This knowledge is foundational in probability theory, providing clarity in numerous real-world scenarios, from predicting races to assessing risks in various fields.

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Most popular questions from this chapter

Give a possible sample space \(\Omega\) for each of the following experiments: (a) An election decides between two candidates \(\mathrm{A}\) and \(\mathrm{B}\). (b) A two-sided coin is tossed. (c) A student is asked for the month of the year and the day of the week on which her birthday falls. (d) A student is chosen at random from a class of ten students. (e) You receive a grade in this course.

If \(A, B,\) and \(C\) are any three events, show that $$ \begin{aligned} P(A \cup B \cup C)=& P(A)+P(B)+P(C) \\ &-P(A \cap B)-P(B \cap C)-P(C \cap A) \\ &+P(A \cap B \cap C) \end{aligned} $$

A die is loaded in such a way that the probability of each face turning up is proportional to the number of dots on that face. (For example, a six is three times as probable as a two.) What is the probability of getting an even number in one throw?

Our calendar has a 400-year cycle. B. H. Brown noticed that the number of times the thirteenth of the month falls on each of the days of the week in the 4800 months of a cycle is as follows: Sunday 687 Monday 685 Tuesday 685 Wednesday 687 Thursday 684 Friday 688 Saturday 684 From this he deduced that the thirteenth was more likely to fall on Friday than on any other day. Explain what he meant by this.

Let \(\Omega=\\{a, b, c\\}\) be a sample space. Let \(m(a)=1 / 2, m(b)=1 / 3,\) and \(m(c)=1 / 6 .\) Find the probabilities for all eight subsets of \(\Omega\).
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