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Chapter 6: Problem 11

Redo exercise 10 assumuing that the first um contains 4 blue balls and 16 white balls and the second urn contains 10 blue balls and 9 white balls.

Short Answer

Expert verified
The probability of picking a blue ball from both urns is \(\frac{10}{95}\).

Step by step solution

01

Understand the problem

We have two urns: Urn 1 contains 4 blue balls and 16 white balls, while Urn 2 contains 10 blue balls and 9 white balls. We will pick 1 ball from Urn 1 (without replacing it) and then 1 ball from Urn 2. We need to find the probability of picking a blue ball from both urns.
02

Calculate the probability of picking a blue ball from Urn 1

In Urn 1, there are a total of 4 blue balls and 16 white balls, making a total of 20 balls. The probability of picking a blue ball from Urn 1 is given by the ratio of blue balls to the total number of balls in the urn: \(P(\text{blue ball from Urn 1}) = \frac{\text{number of blue balls in Urn 1}}{\text{total number of balls in Urn 1}}\) \(P(\text{blue ball from Urn 1}) = \frac{4}{20} = \frac{1}{5}\)
03

Calculate the probability of picking a blue ball from Urn 2

In Urn 2, there are a total of 10 blue balls and 9 white balls, making a total of 19 balls. The probability of picking a blue ball from Urn 2 is given by the ratio of blue balls to the total number of balls in the urn: \(P(\text{blue ball from Urn 2}) = \frac{\text{number of blue balls in Urn 2}}{\text{total number of balls in Urn 2}}\) \(P(\text{blue ball from Urn 2}) = \frac{10}{19}\)
04

Calculate the probability of picking a blue ball from both urns

Since the events are independent, we can find the probability of picking a blue ball from both urns by multiplying their individual probabilities: \(P(\text{blue ball from both urns}) = P(\text{blue ball from Urn 1}) \times P(\text{blue ball from Urn 2})\) \(P(\text{blue ball from both urns}) = \frac{1}{5} \times \frac{10}{19} = \frac{10}{95}\) So the probability of picking a blue ball from both urns is \(\frac{10}{95}\).

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Key Concepts

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

Independent Events
In probability theory, understanding the idea of independent events helps us determine the likelihood of multiple outcomes occurring. Two events are considered independent if the occurrence of one does not affect the occurrence of the other.
For example, when dealing with multiple urns in a probability problem, selecting a ball from one urn typically does not influence the selection from the other.
  • This concept assumes that the chosen items do not interact with or affect each other between selections.
  • As a result, probabilities are calculated separately, with confidence that one event will not change the outcome of the other.
Grasping the independence of events allows us to easily compute the combined probabilities. Just multiply the probability of one event by the probability of the other to get the overall likelihood of both events happening.
Combinatorics
Combinatorics is a branch of mathematics that deals with counting and arranging, laying the groundwork for calculating probabilities. Understanding combinatorics is essential for solving problems involving selections or arrangements from a particular set.
In this context, we are dealing with selecting balls from urns, where each selection is made from a distinct set (the content of each urn).
  • Combinatorics allows us to define how many different ways there are to choose items from a set, often without regard to the order of selection.
  • When calculating probabilities, we often compare the number of favorable outcomes to the total possible outcomes.
This understanding helps build the initial probability framework used in our exercise, shaping our analysis of the scenarios, whether it's picking one ball or considering all possible outcomes in more complex probability problems.
Step-by-step Solutions
Breaking down a problem into step-by-step solutions is a powerful method for tackling complex calculations systematically and with reduced error.
This approach ensures clarity and structure in solving probability exercises like the one involving urns with balls.
  • Start by defining the problem clearly; understand the data and what is asked.
  • Calculate probabilities for each part of the problem separately. For instance, first determine the chance of drawing a blue ball from each individual urn.
  • The final step involves synthesizing these results; for independent events, multiply individual probabilities to find the probability of both occurrences happening together.
Using detailed steps, much like creating a recipe, encourages a deeper understanding and ensures every aspect of the problem is considered, leading to the correct solution by the completion of all steps.

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Most popular questions from this chapter

One urn contains 10 red balls and 25 green balls, and a second urn contains 22 red balls and 15 green balls. A ball is chosen as follows: First an urn is selected by tossing a loaded coin with probability \(0.4\) of landing heads up and probability \(0.6\) of landing tails up. If the coin lands heads up, the first um is chosen; otherwise, the second urn is chosen. Then a ball is picked at random from the chosen um. a. What is the probability that the chosen ball is green? b. If the chosen ball is green, what is the probability that it was picked from the first urn?

a. How many distinguishable ways can the letters of the word \(H U L L A B A L O O\) be arranged? b. How many distinguishable arrangements of the letters of \(H U L L A B A L O O\) begin with \(U\) and end with \(L\) ? c. How many distinguishable arrangements of the letters of \(H U L L A B A L O O\) contain the two letters \(H U\) next to each other in order?

a. How many 3-permutations are there of a set of five objects? b. How many 2-permutations are there of a set of eight objects?

Prove that for all integers \(n \geq 2\), $$ P(n+1,3)=n^{3}-n . $$

A company uses two proofreaders \(X\) and \(Y\) to check a certain manuscript. \(X\) misses \(12 \%\) of typographical errors and \(Y\) misses \(15 \%\). Assume that the proofreaders work independently. a. What is the probability that a randomly chosen typographical error will be missed by both proofreaders? b. If the manuscript contains 1,000 typographical errors, what number can be expected to be missed?
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