/*! 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 21 An experiment consists of random... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 21

An experiment consists of randomly selecting one of three coins, tossing it, and observing the outcome-heads or tails. The first coin is a two-headed coin, the second is a biased coin such that \(P(\mathrm{H})=.75\), and the third is a fair coin. a. What is the probability that the coin that is tossed will show heads? b. If the coin selected shows heads, what is the probability that this coin is the fair coin?

Short Answer

Expert verified
a. The probability that the tossed coin will show heads is \(\frac{4}{6}\). b. If the coin selected shows heads, the probability that this coin is the fair coin is \(\frac{1}{4}\).

Step by step solution

01

Define the Events and Probabilities

We begin by defining the events: A1: Selecting the two-headed coin A2: Selecting the biased coin A3: Selecting the fair coin H: Tossing a coin and getting heads The probabilities of selecting each coin are equal and can be defined as follows: P(A1) = P(A2) = P(A3) = 1/3
02

Find the Probabilities of Getting Heads

We need to find the probabilities of getting heads given that we selected each coin. P(H|A1) = 1 (As this is a two-headed coin) P(H|A2) = 0.75 (As given in the problem) P(H|A3) = 0.5 (As this is a fair coin)
03

a. Calculate the Total Probability of Getting Heads

To find the total probability of getting heads after tossing the coin, we can use the Law of Total Probability. This law states that the probability of an event (H) can be calculated by summing the probabilities of each possible cause (A1, A2, or A3) and the probability of the event (H) given that cause. P(H) = P(H|A1)P(A1) + P(H|A2)P(A2) + P(H|A3)P(A3) Now, substituting the given probabilities, we get: P(H) = (1)(1/3) + (0.75)(1/3) + (0.5)(1/3) P(H) = 1/3 + 1/4 + 1/6 P(H) = 4/6 #Answer_a#: The probability that the tossed coin will show heads is 4/6.
04

b. Calculate the probability that the coin showing heads is the fair coin

Now we need to find the probability that the coin showing heads is the fair coin. For this, we can use Bayes' theorem. Bayes' theorem states that the conditional probability P(A3|H) can be found using the following formula: P(A3|H) = P(H|A3)P(A3) / P(H) We have already calculated P(H) in step a, and we know P(H|A3) and P(A3) from step 1. Now, we substitute these values into the formula: P(A3|H) = (0.5)(1/3) / (4/6) Simplify the expression: P(A3|H) = (0.5)(1/3) / (2/3) P(A3|H) = 1/4 #Answer_b#: If the coin selected shows heads, the probability that this coin is the fair coin is 1/4.

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!

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

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?

The 1992 U.S. Senate was composed of 57 Democrats and 43 Republicans. Of the Democrats, 38 served in the military, whereas 28 of the Republicans had seen military service. If a senator selected at random had served in the military, what is the probability that he or she was Republican?

Applicants for temporary office work at Carter Temporary Help Agency who have successfully completed a typing test are then placed in suitable positions by Nancy Dwyer and Darla Newberg. Employers who hire temporary help through the agency return a card indicating satisfaction or dissatisfaction with the work performance of those hired. From past experience it is known that \(80 \%\) of the employees placed by Nancy are rated as satisfactory, and \(70 \%\) of those placed by Darla are rated as satisfactory. Darla places \(55 \%\) of the temporary office help at the agency and Nancy the remaining \(45 \%\). If a Carter office worker is rated unsatisfactory, what is the probability that he or she was placed by Darla?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a subset of \(B\), then \(P(A) \leq P(B)\).

QuaLrr ConTRoL It is estimated that \(0.80 \%\) of a large consignment of eggs in a certain supermarket is broken. a. What is the probability that a customer who randomly selects a dozen of these cggs receives at least one broken egg? b. What is the probability that a customer who selects these eggs at random will have to check three cartons before finding a carton without any broken eggs? (Each carton contains a dozen eggs.)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

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.