/*! 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 5 Let \(X\) represent the differen... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 5

Let \(X\) represent the difference between the number of heads and the number of tails obtained when a coin is tossed \(n\) times. What are the possible values of \(X ?\)

Short Answer

Expert verified
For even values of \(n\), the possible values of \(X\) are: \[X \in \{ 0, 2, 4, \dots , n \}\] For odd values of \(n\), the possible values of \(X\) are: \[X \in \{ 1, 3, 5, \dots , n \}\]

Step by step solution

01

Identifying the minimum difference

: In the minimum difference case, the number of heads and the number of tails would be equal. This can happen only when we have an even number of tosses (n is even). In this case, we can have half heads and half tails, which makes the minimum difference 0 (zero). If n is odd, it is impossible to have an equal number of heads and tails, so the minimum difference would be 1.
02

Identifying the maximum difference

: The maximum difference between the number of heads and the number of tails would occur when either all tosses result in heads or all tosses result in tails. In this case, X would be equal to the total number of tosses, n.
03

Listing possible values of X

: Now that we know the bounds of X, we can list all possible values within these bounds. For even values of n, the minimum difference in heads and tails is 0, and the maximum difference is n. For odd values of n, the minimum difference is 1, and the maximum difference is still n. In each case, the difference will always be an even number. So, for an even value of n, the possible values of X are: \[x \in \{ 0, 2, 4, \dots , n \}\] For an odd value of n, the possible values of X are: \[x \in \{ 1, 3, 5, \dots , n \}\]

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

Suppose that a die is rolled twice. What are the possible values that the following random variables can take on: (a) the maximum value to appear in the two rolls; (b) the minimum value to appear in the two rolls; (c) the sum of the two rolls; (d) the value of the first roll minus the value of the second roll?

A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win \(\$ 1.10 ;\) if they are different colors, then you win \(-\$ 1.00 .\) (That is, you lose \(\$ 1.00 .\) ) Calculate (a) the expected value of the amount you win; (b) the variance of the amount you win.

Suppose that 10 balls are put into 5 boxes, with each ball independently being put in box \(i\) with probability \(p_{i}, \sum_{i=1}^{5} p_{i}=1\) (a) Find the expected number of boxes that do not have any balls. (b) Find the expected number of boxes that have exactly 1 ball.

\(A\) and \(B\) play the following game: \(A\) writes down either number 1 or number \(2,\) and \(B\) must guess which one. If the number that \(A\) has written down is \(i\) and \(B\) has guessed correctly, \(B\) receives \(i\) units from \(A .\) If \(B\) makes a wrong guess, \(B\) pays \(\frac{3}{4}\) unit to A. If \(B\) randomizes his decision by guessing 1 with probability \(p\) and 2 with probability \(1-p,\) determine his expected gain if (a) \(A\) has written down number 1 and (b) \(A\) has written down number 2 What value of \(p\) maximizes the minimum possible value of \(B^{\prime}\) s expected gain, and what is this maximin value? (Note that \(B\) 's expected gain depends not only on \(p,\) but also on what \(A\) does. Consider now player \(A .\) Suppose that she also randomizes her decision, writing down number 1 with probability \(q\). What is \(A\) 's expected loss if (c) \(B\) chooses number 1 and \((\text { d) } B \text { chooses number } 2 ?\) What value of \(q\) minimizes \(A\) 's maximum expected loss? Show that the minimum of \(A\) 's maximum expected loss is equal to the maximum of \(B\) 's minimum expected gain. This result, known as the minimax theorem, was first established in generality by the mathematician John von Neumann and is the fundamental result in the mathematical discipline known as the theory of games. The common value is called the value of the game to player \(B\).

Each of 500 soldiers in an army company independently has a certain disease with probability \(1 / 10^{3}\). This disease will show up in a blood test, and to facilitate matters, blood samples from all 500 soldiers are pooled and tested. (a) What is the (approximate) probability that the blood test will be positive (that is, at least one person has the disease)? Suppose now that the blood test yields a positive result. (b) What is the probability, under this circumstance, that more than one person has the disease? One of the 500 people is Jones, who knows that he has the disease. (c) What does Jones think is the probability that more than one person has the disease? Because the pooled test was positive, the authorities have decided to test each individual separately. The first \(i-1\) of these tests were negative, and the \(i\) th one-which was on Jones-was positive. (d) Given the preceding, scenario, what is the probability, as a function of \(i,\) that any of the remaining people have the disease?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.