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Ch. 4 - Probability
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
Chapter 4, Problem 4.4.7

Quinela In a horse race, a quinela bet is won if you selected the two horses that finish first and second, and they can be selected in any order. The 144th running of the Kentucky Derby had a field of 20 horses. What is the probability of winning a quinela bet if you randomly select the horses?

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Step 1: Understand the problem. A quinela bet involves selecting two horses that finish first and second in any order. The total number of horses in the race is 20.
Step 2: Calculate the total number of ways to choose 2 horses from a group of 20. This is a combination problem, as the order of selection does not matter. Use the combination formula: C(n, k) = n! / [k!(n-k)!], where n is the total number of items (20 horses) and k is the number of items to choose (2 horses).
Step 3: Substitute the values into the combination formula: C(20, 2) = 20! / [2!(20-2)!]. Simplify the factorials to compute the total number of combinations.
Step 4: Since you are randomly selecting 2 horses, there is only 1 favorable outcome (the pair of horses you selected finishing first and second in any order). The probability of winning the quinela bet is the ratio of favorable outcomes to total outcomes: P = 1 / C(20, 2).
Step 5: Simplify the probability expression to get the final probability. This involves calculating the denominator (C(20, 2)) and then taking its reciprocal.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and arrangements of objects. In the context of the quinela bet, it helps determine how many ways two horses can be selected from a group of 20. The formula for combinations is used to calculate the number of ways to choose 'r' items from 'n' items without regard to the order of selection.

Probability

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In this scenario, the probability of winning a quinela bet can be calculated by dividing the number of successful outcomes (correctly selecting the first and second horses) by the total number of possible outcomes (all combinations of two horses from the field).
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Permutations vs. Combinations

Permutations and combinations are both ways to count arrangements of items, but they differ in whether order matters. In a quinela bet, the order of the selected horses does not matter, making it a combination problem. Understanding this distinction is crucial for correctly calculating the probability of winning the bet.
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Related Practice
Textbook Question

In Exercises 9–20, use the data in the following table, which lists survey results from high school drivers at least 16 years of age (based on data from “Texting While Driving and Other Risky Motor Vehicle Behaviors Among U.S. High School Students,” by O’Malley, Shults, and Eaton, Pediatrics, Vol. 131, No. 6). Assume that subjects are randomly selected from those included in the table. Hint: Be very careful to read the question correctly.

Texting and Alcohol If one of the high school drivers is randomly selected, find the probability that the selected driver did not text while driving and did not drive when drinking.

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Textbook Question

In Exercises 33–40, use the given probability value to determine whether the sample results are significant.



Voting Repeat the preceding Exercise 33 after replacing 40 Democrats being placed on the first line of voting ballots with 26 Democrats being placed on the first line. The probability of getting a result as high as 26 is 0.058638.

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Textbook Question

In Exercises 13–20, express the indicated degree of likelihood as a probability value between 0 and 1.



Movies Based on a study of the movies made in a recent year, 33 out of every 100 movies have a female lead or co-lead.

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Textbook Question

In Exercises 29–32, use the given sample space or construct the required sample space to find the indicated probability.



Four Children Exercise 29 lists the sample space for a couple having three children. After identifying the sample space for a couple having four children, find the probability of getting three girls and one boy (in any order).

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Textbook Question

Teed Off When four golfers are about to begin a game, they often toss a tee to randomly select the order in which they tee off. What is the probability that they tee off in alphabetical order by last name?

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Textbook Question

Mendel’s Peas Mendel conducted some of his famous experiments with peas that were either smooth yellow plants or wrinkly green plants. If four peas are randomly selected from a batch consisting of four smooth yellow plants and four wrinkly green plants, find the probability that the four selected peas are of the same type.

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