91Ó°ÊÓ

Based on the following list oftop \(D V D\) rentals (based on revenue) for the weekend ending January 4, 2009:$$\begin{array}{|l|c|}\hline \text { Name } & \text { Rental Index } \\\\\hline \text { EagleEye } & 100.00 \\\\\hline \text { Burn After Reading } & 74.62 \\\\\hline \text { Mamma Mia! } &63.30\\\\\hline \text { The Dark Knight } & 62.43 \\\\\hline \text { Death Race } & 61.50 \\\\\hline\begin{array}{l}\text { The Mummy: Tomb of the } \\\\\text { Dragon Emperor }\end{array} & 60.72 \\\\\hline \text { Traitor } & 52.57 \\\\\hline \text { Wanted } & 49.22 \\\\\hline \text { Step Brothers } & 46.81 \\\\\hline \text { Horton Hears a Who! } & 43.91 \\\\\hline\end{array}$$ Rather than study for astrophysics, you and your friends decide to get together for a marathon movie-watching gummybear-munching event on Saturday night. You decide to watch three movies selected at random from the above list. a. How many sets of three movies are possible? b. Your best friends, the Pelogrande twins, refuse to see either Mamma Mia! or The Mummy on the grounds that they are "for idiots" and also insist that no more than one of Traitor and Death Race should be among the movies selected. How many of the possible groups of three will satisfy the twins? c. Comparing the answers in parts (a) and (b), would you say the Pelogrande twins are more likely than not to be satisfied with your random selection?

Step by step solution

01

Part (a) - Calculating total movie sets.

To calculate the total number of movie sets, we need to find the number of combinations we can make using the 10 movies while choosing 3 at a time. This can be done using the formula for combinations: \[C(n, k) = \frac{n!}{k!(n-k)!}\], where n is the number of items in the set (10 movies) and k is the number of items we want to choose (3 movies). Using the formula, we get: \[C(10, 3) = \frac{10!}{3!(10-3)!} = \frac{10!}{3!7!}\]

02

Calculate the factorials

Now we proceed to calculate the factorials: \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\) \(3! = 3 \times 2 \times 1\) \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)

03

Plug in factorials

Plugging the factorials into the formula, we get: \(C(10, 3) = \frac{10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{(3\times2\times1)(7\times6\times5\times4\times3\times2\times1)}\)

04

Simplify the expression

Simplifying the expression, we get: \(C(10, 3) = \frac{10\times9\times8}{3\times2\times1} = 120\) So there are 120 sets of three movies possible.

05

Part (b) - Calculate the sets satisfying the twins' conditions

There are two conditions given by the Pelogrande twins: 1. They don't want Mamma Mia! or The Mummy in the selection. 2. No more than one of Traitor and Death Race should be selected. Let's first remove Mamma Mia! and The Mummy from the list, leaving us with 8 movies remaining. Of these 8 movies, we can select any 3 freely for the first two conditions. So the number of combinations now is: \[C(8, 3) = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!}\] Calculating factorials and simplifying the expression: C(8,3) = \(\frac{8\times7\times6}{3\times2\times1} = 56\) So there are 56 sets of three movies that will satisfy the first two conditions.

06

Part (c) - Analyze the twins' satisfaction with the selection

Now we have: - Total possible sets of three movies: 120 sets - Sets of three movies that satisfy the twins' conditions: 56 sets To find the probability that the twins will be satisfied with the random selection, we divide the number of satisfying sets by the total number of possible sets: \[P(\text{Satisfied}) = \frac{\text{Number of satisfying sets}}{\text{Total number of possible sets}} = \frac{56}{120}\] Simplifying the fraction, we get: \[P(\text{Satisfied}) = \frac{14}{30} = 0.4667\] Since the probability is less than 0.5 (50%), the Pelogrande twins are less likely to be satisfied with the random selection.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Combinations

In the world of mathematics, particularly in combinatorics, combinations play a crucial role in determining how we can select items from a larger pool without regard to the order of selection. When your problem involves choosing a certain number of items from a larger set and the order of selection doesn't matter, you're dealing with combinations.

To figure out how many combinations are possible, we use the formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where:

• \( n \) is the total number of items,
• \( k \) is the number of items to choose,
• \(!\) denotes factorial, a product of all positive integers up to a specified number.

In the exercise, this formula helps determine how many sets of three movies can be formed from a list of 10 movies. By calculating \(C(10, 3)\), we found that there are 120 possible sets.

Understanding the concept of combinations is essential for dealing with any problems where the selection of subsets is required without repetition of items.

Probability

Probability is the mathematical way of expressing the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

In this exercise, probability helps us understand how likely it is for a set of movies to be chosen that satisfies the Pelogrande twins' requirements. To find the probability that the twins will be satisfied, we calculate: \[ P(\text{Satisfied}) = \frac{\text{Number of Satisfaction Sets}}{\text{Total Possible Sets}} \]Executing this calculation in the context of our exercise, we found that 56 of the possible sets meet the twins' criteria, out of 120 total possible sets. Thus: \[ P(\text{Satisfied}) = \frac{56}{120} \approx 0.4667 \] This equals roughly 46.67%, indicating that the likelihood of randomly choosing a satisfactory movie set is less than half. Understanding probability is key to making informed predictions about how often a particular event will occur given certain conditions.

Factorial

Factorials are a mathematical operation that is fundamental in combinatorics and probability. They involve the multiplication of a series of descending natural numbers. The factorial of a non-negative integer \( n \), denoted \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \). Factorials grow very fast as the number increases, making them suitable for calculations involving permutations and combinations.

In the context of the exercise, factorials are used within the formula for combinations. The calculation of \(C(10, 3)\) involves factorials of 10, 3, and 7, computed as:

• \(10!\)
• \(3!\)
• \(7!\)

By breaking down each factorial into a series of multiplications, we simplify the equation to determine the number of possible combinations. This makes factorials a vital tool in solving problems that demand the selection of items from a group without replacement and helps clarify the calculation of probabilities in such scenarios. Understanding how to compute and use factorials is essential for mastering more advanced topics in statistics and probability.