Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 5: Q4E (page 337)

Suppose that a die is loaded so that each of the numbers 1, 2, 3, 4, 5, and 6 has a different probability of appearing when the die is rolled. For\({\bf{i = 1, \ldots ,6,}}\)let\({{\bf{p}}_{\bf{i}}}\)denote the probability that the number i will be obtained, and

suppose that\({{\bf{p}}_{\bf{1}}}{\bf{ = 0}}{\bf{.11,}}\,\,{{\bf{p}}_{\bf{2}}}{\bf{ = 0}}{\bf{.30,}}\,\,{{\bf{p}}_{\bf{3}}}{\bf{ = 0}}{\bf{.22,}}\,\,{{\bf{p}}_{\bf{4}}}{\bf{ = 0}}{\bf{.05,}}\,\,{{\bf{p}}_{\bf{5}}}{\bf{ = 0}}{\bf{.25}}\,\,{\bf{and}}\,{{\bf{p}}_{\bf{6}}}{\bf{ = 0}}{\bf{.07}}{\bf{.}}\)Suppose also that the die is to be rolled 40 times. Let\({{\bf{X}}_{\bf{1}}}\)denote the number of rolls for which an even number appears, and let\({{\bf{X}}_2}\)denote the number of rolls for which either the number 1 or the number 3 appears. Find the value of\({\bf{Pr}}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ = 20}}\,\,{\bf{and}}\,\,{{\bf{X}}_{\bf{2}}}{\bf{ = 15}}} \right)\).

Short Answer

Expert verified

\(8.0727 \times {10^{ - 3}}\)

Step by step solution

01

Given information

It is given that \({p_1} = 0.11,\,\,{p_2} = 0.30,\,\,{p_3} = 0.22,\,\,{p_4} = 0.05,\,\,{p_5} = 0.25\,\,and\,{p_6} = 0.07.\)These are the probability values of different numbers that appear on a die, that is , 1,2,3,4,5 and 6.

The die is rolled 40 times

Here two events are described:

\({X_1}\): Denotes the number of rolls for which an even number appears

\({X_2}\): Denotes the event when either number 1 or 3 occurs.

02

Define the pdf

The probability that on a roll an even number occurs is

\(\begin{array}{l} = {p_2} + {p_4} + {p_6}\\ = 0.3 + 0.05 + 0.07\\ = 0.42\end{array}\)

Therefore, the distribution is

\({X_1} \sim Bin\left( {n = 40,p = 0.42} \right)\)

The probability that on a roll an either 1 or 3 occurs is

\(\begin{array}{l} = {p_1} + {p_3}\\ = 0.11 + 0.22\\ = 0.33\end{array}\)

Therefore, the distribution is

\({X_2} \sim Bin\left( {n = 40,p = 0.33} \right)\)

03

Calculate the desired probability

We need to find:

\(\begin{array}{l}\Pr \left( {{X_1} = 20\,\,and\,\,{X_2} = 15} \right)\\ = {}^{40}{C_{20}}{\left( {0.42} \right)^{20}}{\left( {0.58} \right)^{20}} \times {}^{40}{C_{15}}{\left( {0.33} \right)^{15}}{\left( {0.67} \right)^{25}}\\ = 8.0727 \times {10^{ - 3}}\end{array}\)

Hence the required solution is\({\bf{8}}{\bf{.0727 \times 1}}{{\bf{0}}^{{\bf{ - 3}}}}\)

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

Suppose that a box contains five red balls and ten blue balls. If seven balls are selected randomly without replacement, what is the probability that at least three red balls will be obtained?

Suppose that 15,000 people in a city with a population of 500,000 are watching a certain television program. If 200 people in the city are contacted at random, what is the approximate probability that fewer than four of themare watching the program?

Suppose that the random variables \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{k}}}\) are independent and that \({{\bf{X}}_{\bf{i}}}\) has the Poisson distribution with mean \({{\bf{\lambda }}_{\bf{i}}}\left( {{\bf{i = 1, \ldots ,k}}} \right)\). Show that for each fixed positive integer n, the conditional distribution of the random Vector \({\bf{X = }}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{k}}}} \right)\), given that \(\sum\limits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{X}}_{\bf{i}}}{\bf{ = n}}} \) it is the multinomial distribution with parameters n and

\(\begin{array}{l}{\bf{p = }}\left( {{{\bf{p}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{p}}_{\bf{k}}}} \right){\bf{,}}\,{\bf{where}}\\{{\bf{p}}_{\bf{i}}}{\bf{ = }}\frac{{{{\bf{\lambda }}_{\bf{i}}}}}{{\sum\limits_{{\bf{j = 1}}}^{\bf{k}} {{{\bf{\lambda }}_{\bf{j}}}} }}\,{\bf{for}}\,{\bf{i = 1, \ldots ,k}}{\bf{.}}\end{array}\)

Suppose that X has the normal distribution with mean\({\bf{\mu }}\)and variance\({{\bf{\sigma }}^{\bf{2}}}\). Express\({\bf{E}}\left( {{{\bf{X}}^{\bf{3}}}} \right)\)in terms of\({\bf{\mu }}\)and\({{\bf{\sigma }}^{\bf{2}}}\).

Suppose that seven balls are selected at random withoutreplacement from a box containing five red balls and ten blue balls. If \(\overline X \) denotes the proportion of red balls in the sample, what are the mean and the variance of \(\overline X \) ?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

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