/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 Find the probability of selectin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 12

Find the probability of selecting 3 science books and 4 math books from 8 science books and 9 math books. The books are selected at random.

Short Answer

Expert verified
The probability is approximately 0.182 or \( \frac{102}{560} \) when simplified.

Step by step solution

01

Understanding the Problem

We need to find the probability of selecting 3 science books from a total of 8 science books and 4 math books from a total of 9 math books. This involves calculating combinations for selecting these books and then determining the probability from these combinations.
02

Find Total Ways to Select Books

First, determine the total number of ways to select any 7 books from the total of 17 books (8 science and 9 math books). This is calculated using the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, \( n = 17 \) and \( r = 7 \), so we compute \( \binom{17}{7} \).
03

Calculate Combinations of Science Books

Calculate the number of ways to choose 3 science books from 8 available science books. This is given by \( \binom{8}{3} = \frac{8!}{3!(8-3)!} \).
04

Calculate Combinations of Math Books

Calculate the number of ways to choose 4 math books from 9 available math books. This is given by \( \binom{9}{4} = \frac{9!}{4!(9-4)!} \).
05

Calculate Favorable Outcomes

Multiply the combinations of selecting 3 science books by the combinations of selecting 4 math books. These are the favorable outcomes: \( \binom{8}{3} \times \binom{9}{4} \).
06

Calculate the Probability

The probability of selecting the desired combination of books is the ratio of favorable outcomes to the total number of ways to select 7 books, which is \( \frac{\binom{8}{3} \times \binom{9}{4}}{\binom{17}{7}} \).
07

Perform Calculations

Compute the required combinations: \( \binom{8}{3} = 56 \), \( \binom{9}{4} = 126 \), and \( \binom{17}{7} = 19448 \). Then, calculate the probability: \( \frac{56 \times 126}{19448} \). Simplify the result to find the final probability.
08

Present the Result

After calculations, the probability of selecting 3 science books and 4 math books is approximately \( 0.182 \) or \( \frac{102}{560} \) when simplified.

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 91影视!

Key Concepts

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

Probability calculation
Probability gives us a value representing the likelihood of an event occurring. To compute this for a set of choices, such as selecting books, involves comparing favorable outcomes to the total possible outcomes.
The formula for probability is:\[P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\] This problem asks us to calculate the probability of selecting certain numbers of books from a mixed collection.
Thus, our task involves counting the number of successful book-picking combinations and dividing by the total combinations possible.
Understanding these concepts simplifies how complex probability problems, like the one concerning multiple book selections, are tackled.
Combination formula
The combination formula is a cornerstone of combinatorics, allowing us to count combinations of items where order does not matter.
This formula is:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\] Where:
  • \(n\) is the total number of items,
  • \(r\) is the number of items to choose,
  • \(!\) denotes factorial, the product of all positive integers up to that number.
Using this formula, we can calculate various selections: in our problem, selecting 3 books from 8 and selecting 4 from 9.
This method of calculating enables us to handle larger numbers efficiently and ensures no repetitive hand-counting, making it crucial for solving probability issues.
With a solid grasp on this formula, you can tackle numerous real-world problems involving selections.
Favorable outcomes
In probability, 'favorable outcomes' are those specific cases that meet the criteria we are interested in. To solve any probability problem, identifying and counting these favorable outcomes is essential.
In the book selection problem, the favorable outcomes would involve selecting exactly 3 science books and 4 math books. This is because these are the exact criteria we want to meet.
To find the number of favorable outcomes, you multiply the combinations of selecting the specific science books by the combinations of selecting math books that satisfy the requirement.
  • Combinations for science: \(\binom{8}{3}\)
  • Combinations for math: \(\binom{9}{4}\)
  • Favorable outcomes: \(\binom{8}{3} \times \binom{9}{4}\)
By understanding this concept, you can breakdown larger probability problems into manageable calculations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Toss two coins 100 times and record the number of heads (0,1,2) . Compute the probabilities of each outcome, and compare these probabilities with the theoretical results.

The American Automobile Association (AAA) reports that of the fatal car and truck accidents, \(54 \%\) are caused by car driver error. If 3 accidents are chosen at random, find the probability that a. All are caused by car driver error b. None is caused by car driver error c. At least 1 is caused by car driver error

Odds are used in gambling games to make them fair. For example, if you rolled a die and won every time you rolled a \(6,\) then you would win on average once every 6 times. So that the game is fair, the odds of 5 to 1 are given. This means that if you bet \(\$ 1\) and won, you could win \(\$ 5 .\) On average, you would win \(\$ 5\) once in 6 rolls and lose \(\$ 1\) on the other 5 rolls - hence the term fair game. In most gambling games, the odds given are not fair. For example, if the odds of winning are really 20 to 1 the house might offer 15 to 1 in order to make a profit. Odds can be expressed as a fraction or as a ratio, such as \(\frac{5}{1}, 5: 1,\) or 5 to \(1 .\) Odds are computed in favor of the event or against the event. The formulas for odds are $$ \begin{array}{l} \text { Odds in favor }=\frac{P(E)}{1-P(E)} \\ \text { Odds against }=\frac{P(\bar{E})}{1-P(\bar{E})} \end{array} $$ In the die example, $$ \begin{aligned} &\text { Odds in favor of a } 6=\frac{\frac{1}{6}}{\frac{5}{6}}=\frac{1}{5} \text { or } 1: 5\\\ &\text { Odds against a } 6=\frac{\frac{5}{6}}{\frac{1}{6}}=\frac{5}{1} \text { or } 5: \end{aligned} $$ Find the odds in favor of and against each event. a. Rolling a die and getting a 2 b. Rolling a die and getting an even number c. Drawing a card from a deck and getting a spade d. Drawing a card and getting a red card e. Drawing a card and getting a queen f. Tossing two coins and getting two tails g. Tossing two coins and getting exactly one tail

How many ways can you pick 4 students from 10 students \((6\) men, 4 women ) if you must have an equal number of each gender or all of the same gender?

An advertising manager decides to have an ad campaign in which 8 special calculators will be hidden at various locations in a shopping mall. If he has 17 locations from which to pick, how many different possible combinations can he choose?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.