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Chapter 7: Problem 10

Four balls are selected at random without replacement from an urn containing three white balls and five blue balls. Find the probability of the given event. All of the balls are blue.

Short Answer

Expert verified
The probability of selecting all four blue balls from the urn is \( \frac{5}{56} \).

Step by step solution

01

Identify the total number of ways to select four balls from the urn

We have three white balls and five blue balls, making a total of eight balls in the urn. We want to find the number of ways to select four balls from these eight balls. This can be calculated using combinations, denoted as C(n, r), where n is the total number of objects and r is the number of objects we want to choose. Here, n = 8 (total balls in urn) and r = 4 (balls to be selected). Total number of ways to select 4 balls from 8 balls will be: C(8, 4) = \( \frac{8!}{4! (8-4)!} \)
02

Calculate the number of ways to select 4 blue balls

Now, we will find the number of ways to select all four blue balls from the five available blue balls in the urn. Using combinations: n = 5 (total blue balls in urn) and r = 4 (blue balls to be selected). Number of ways to select 4 blue balls from 5 blue balls will be: C(5, 4) = \( \frac{5!}{4! (5-4)!} \)
03

Calculate the probability of selecting 4 blue balls

Now, we will divide the number of successful cases (selecting 4 blue balls) by the total number of cases (selecting any four balls) to find the desired probability: Probability = \( \frac{\text{Number of ways to select 4 blue balls}}{\text{Total number of ways to select 4 balls}} \) = \( \frac{C(5, 4)}{C(8, 4)} \) = \( \frac{\frac{5!}{4! (5-4)!}}{\frac{8!}{4! (8-4)!}} \) = \( \frac{5}{\frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1}} \) = \( \frac{5 \times 4 \times 3 \times 2 \times 1}{8 \times 7 \times 6 \times 5} \) = \( \frac{5}{56} \) So, the probability of selecting all blue balls is \( \frac{5}{56} \).

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Key Concepts

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

Combinatorics
Combinatorics is a fundamental area of mathematics focused on counting and arranging objects. It's a powerful tool to calculate possibilities in various scenarios, such as forming teams or selecting items from a group. One of its key aspects is understanding combinations.
  • A combination is a selection of items without considering the order. For example, choosing 3 teams from 5 is a combination problem where the order doesn't matter.
  • Mathematically, combinations are represented using the formula: \( C(n, r) = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items and \( r \) is the number of items to be selected.
  • In our exercise, this principle helps us calculate how many ways we can select various numbers of balls from a set of different-colored balls.
Understanding combinatorics is crucial in probability, as it helps determine the number of favorable and total outcomes.
Random Selection
Random selection plays a pivotal role in probability theory. It refers to choosing items from a set where each item has an equal chance of being picked. In real-world applications, this might be like drawing names out of a hat.
  • In many scenarios, such as our exercise with selecting balls, random selection implies that each possible selection has the same likelihood.
  • This ensures fairness and an unbiased outcome, which is vital in experiments and statistics.
  • For instance, if you randomly select four balls from an urn containing different-colored balls, each combination of four has an equal chance of occurring. By understanding and applying random selection, we can derive meaningful probabilities and make accurate predictions about outcomes.
Finite Mathematics
Finite mathematics includes various mathematical principles applied in situations with clearly defined limits or boundaries. It covers a range of topics like counting methods, probability, and finite sets.
  • In finite mathematics, problems often deal with fixed sets, like our urn with a specific number of balls.
  • Methods from finite mathematics, such as combinations and permutations, allow us to organize and analyze limited data effectively.
  • The fixed nature of finite problems, such as knowing there are only 8 balls in total, makes it easier to find solutions using concise mathematical strategies and tools.
Mastery of finite mathematics is essential for solving real-world problems, as it combines logic and quantitative reasoning.

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Most popular questions from this chapter

The chief loan officer of La Crosse Home Mortgage Company summarized the housing loans extended by the company in 2007 according to type and term of the loan. Her list shows that \(70 \%\) of the loans were fixed-rate mortgages \((F), 25 \%\) were adjustable-rate mortgages \((A)\), and \(5 \%\) belong to some other category \((O)\) (mostly second trust-deed loans and loans extended under the graduated payment plan). Of the fixed-rate mortgages, \(80 \%\) were 30 -yr loans and \(20 \%\) were 15 -yr loans; of the adjustable-rate mortgages, \(40 \%\) were 30 -yr loans and \(60 \%\) were 15 -yr loans; finally, of the other loans extended, \(30 \%\) were 20 -yr loans, \(60 \%\) were 10 -yr loans, and \(10 \%\) were for a term of 5 yr or less. a. Draw a tree diagram representing these data. b. What is the probability that a home loan extended by La Crosse has an adjustable rate and is for a term of 15 yr? c. What is the probability that a home loan cxtended by La Crosse is for a term of 15 yr?

In a survey on consumer-spending methods conducted in 2006, the following results were obtained: $$\begin{array}{lccccc} \hline & & & & {\text { Debit/ATM }} & \\ \text { Payment Method } & \text { Checks } & \text { Cash } & \text { Credit cards } & \text { cards } & \text { Other } \\ \hline \text { Transactions, \% } & 37 & 14 & 25 & 15 & 9 \\ \hline \end{array}$$ If a transaction tracked in this survey is selected at random, what is the probability that the transaction was paid for a. With a credit card or with a debit/ATM card? b. With cash or some method other than with a check, a credit card, or a debit/ATM card?

The admissions office of a private university released the following admission data for the preceding academic year: From a pool of 3900 male applicants, \(40 \%\) were accepted by the university and \(40 \%\) of these subsequently enrolled. Additionally, from a pool of 3600 female applicants, \(45 \%\) were accepted by the university and \(40 \%\) of these subsequently enrolled. What is the probability that a. A male applicant will be accepted by and subsequently will enroll in the university? b. A student who applies for admissions will he accepted by the university? c. A student who applies for admission will be accepted by the university and subsequently will enroll?

According to a study conducted in 2003 concerning the participation, by age, of \(401(\mathrm{k})\) investors, the following data were obtained: $$ \begin{array}{lccccc} \hline \text { Age } & 20 \mathrm{~s} & 30 \mathrm{~s} & 40 \mathrm{~s} & 50 \mathrm{~s} & 60 \mathrm{~s} \\ \hline \text { Percent } & 11 & 28 & 32 & 22 & 7 \\ \hline \end{array} $$ a. What is the probability that a \(401(\mathrm{k})\) investor selected at random is in his or her 20 s or 60 s? b. What is the probability that a \(401(\mathrm{k})\) investor selected at random is under the age of 50 ?

In a study of the scientific research on soft drinks, juices, and milk, 50 studies were fully sponsored by the food industry, and 30 studies were conducted with no corporate ties. Of those that were fully sponsored by the food industry, \(14 \%\) of the participants found the products unfavorable, \(23 \%\) were neutral, and \(63 \%\) found the products favorable. Of those that had no industry funding, \(38 \%\) found the products unfavorable, \(15 \%\) were neutral, and \(47 \%\) found the products favorable. a. What is the probability that a participant selected at random found the products favorable? b. If a participant selected at random found the product favorable, what is the probability that he or she belongs to a group that participated in a corporate-sponsored study?
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