Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Q. 3.16 (page 107)

Independent trials that result in a success with probability pand a failure with probability 1-pare called Bernoulli trials. Let Pndenote the probability that n Bernoulli trials result in an even number of successes (0 being considered an even number). Show that

Pn=p1-Pn-1+(1-p)Pn-1n≥1

Short Answer

Expert verified

Condition upon the outcome of the first trial. Independence make the following trials, n-1Bernoulli trials.

For the step of the induction substitute explicit formula for Pninto recursive formula.

Step by step solution

01

conditions for the problems

Conditioning to the first trial can solve the problem.

If and only if one of the two disjoint situations below is true, we will get an even number of successes:

  1. The first trial was a success, and in n-1trials, an odd number of successes occurs.
  2. The first trial ended in failure, and n-1trials result in an even number of successes.
02

Independence between the trials

Because of the recursive relationship, the trials are completely independent of one another. in other words,

Pn=p1-Pn-1+qPn-1,∀n≥1

n=1is included since we anticipated 0to be an even integer, hence P0=1.

03

Derivation by Induction

In the other hand, the demonstration by induction goes like this,

n=1,1+1-2p2=q=P1

n, For a universal natural number n, we suppose that the equality holds.

n+1, SubstitutingPn=1+(1-2p)n2

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

A total of 46percent of the voters in a certain city classify themselves as Independents, whereas 30percent classify themselves as Liberals and 24percent say that they are Conservatives. In a recent local election, 35percent of the Independents, 62percent of the Liberals, and 58percent of the Conservatives voted. A voter is chosen at random. Given that this person voted in the local election, what is the probability that he or she is

(a) an Independent?

(b) a Liberal?

(c) a Conservative?

(d) What percent of voters participated in the local election?

Suppose that each child born to a couple is equally likely to be a boy or a girl, independently of the sex distribution of the other children in the family. For a couple having 5children, compute the probabilities of the following events:

(a) All children are of the same sex.

(b) The 3eldest are boys and the others girls.

(c) Exactly 3are boys.

(d) The 2oldest are girls.

(e) There is at least 1girl.

In any given year, a male automobile policyholder will make a claim with probability pmand a female policyholder will make a claim with probability localid="1646823185045" pf,where pf≠pm. The fraction of the policyholders that are male is α,0<α<1.A policyholder is randomly chosen. If Aidenotes the event that this policyholder will make a claim in the year i,show that

PA2∣A1>PA1

Give an intuitive explanation of why the preceding inequality is true.

A total of 48 percent of the women and 37 percent of the men who took a certain “quit smoking” class remained nonsmokers for at least one year after completing the class. These people then attended a success party at the end of a year. If 62 percent of the original class was male,

(a) what percentage of those attending the party were women?

(b) what percentage of the original class attended the party?

Use Equation (2.1)to compute in a hand of bridge the conditional probability that East has 3spades given that North and South have a combined total of 8 spades.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Decision 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.

Sign-up for free