My Math Forum  

Go Back   My Math Forum > High School Math Forum > Probability and Statistics

Probability and Statistics Basic Probability and Statistics Math Forum


Thanks Tree2Thanks
  • 1 Post By romsek
  • 1 Post By JeffM1
Reply
 
LinkBack Thread Tools Display Modes
February 25th, 2017, 07:10 PM   #1
Newbie
 
Joined: Jan 2017
From: Malaysia

Posts: 20
Thanks: 3

Math Focus: calculus
Probability finding "n"

Hi everyone. I have problem regarding solving this questions. Hope some one can help.

Q: For a new express delivery company, the probability of making a late delivery per day is one in twenty. Calculate the maximum number of deliveries per day if the probability of

(a) at least one late delivery is to be less than 0.8
(b) no late delivery is to be more than 0.1
[ans: a=31, b=44]
palongze is offline  
 
February 25th, 2017, 07:37 PM   #2
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: Southern California, USA

Posts: 1,604
Thanks: 817

Quote:
Originally Posted by palongze View Post
Hi everyone. I have problem regarding solving this questions. Hope some one can help.

Q: For a new express delivery company, the probability of making a late delivery per day is one in twenty. Calculate the maximum number of deliveries per day if the probability of

(a) at least one late delivery is to be less than 0.8
(b) no late delivery is to be more than 0.1
[ans: a=31, b=44]
This is just the binomial probability distribution.

$p=\dfrac {1}{20}$

$P[\text{k late out of n deliveries}] = \binom{n}{k}p^k (1-p)^{n-k}$

a) $P[\text{at least 1 late delivery}]=1-P[\text{no late deliveries}]$

$P[\text{no late deliveries}] = (1-p)^n$

$P[\text{at least 1 late delivery}]=1 - (1-p)^n < 0.8$

This is easily solved to obtain $n=31$

b) $(1-p)^n > 0.1$

again easily solved to obtain $n=44$
Thanks from palongze

Last edited by romsek; February 25th, 2017 at 08:32 PM.
romsek is online now  
February 25th, 2017, 07:43 PM   #3
Senior Member
 
Joined: May 2016
From: USA

Posts: 825
Thanks: 335

$a = P(at\ least\ one\ late\ delivery) < 0.8.$

$\therefore b = P(no\ late\ delivery) = 1 - a \implies b > 0.2.$

$b = \dbinom{n}{0} * 0.95^n * 0.05^{(n - n)} = 0.95^n.$

$0.95^{max(n)} > 0.2 \implies max(n)* ln(0.95) > ln(0.2) \implies max(n) > \dfrac{ln(0.2)}{ln(0.95)} > 31 \implies$

$max(n) = 31.$

Let's check that answer.

$1 - 0.95^{31} \approx 0.796 < 0.8,\ but\ 1 - 0.95^{32} \approx 0.806 > .8.$

Do you follow that?
Thanks from palongze
JeffM1 is offline  
February 25th, 2017, 08:56 PM   #4
Newbie
 
Joined: Jan 2017
From: Malaysia

Posts: 20
Thanks: 3

Math Focus: calculus
Quote:
Originally Posted by romsek View Post
This is just the binomial probability distribution.

$p=\dfrac {1}{20}$

$P[\text{k late out of n deliveries}] = \binom{n}{k}p^k (1-p)^{n-k}$

a) $P[\text{at least 1 late delivery}]=1-P[\text{no late deliveries}]$

$P[\text{no late deliveries}] = (1-p)^n$

$P[\text{at least 1 late delivery}]=1 - (1-p)^n < 0.8$

This is easily solved to obtain $n=31$

b) $(1-p)^n > 0.1$

again easily solved to obtain $n=44$
Thank you. I understand where my mistake now.


Quote:
Originally Posted by JeffM1 View Post
$a = P(at\ least\ one\ late\ delivery) < 0.8.$

$\therefore b = P(no\ late\ delivery) = 1 - a \implies b > 0.2.$

$b = \dbinom{n}{0} * 0.95^n * 0.05^{(n - n)} = 0.95^n.$

$0.95^{max(n)} > 0.2 \implies max(n)* ln(0.95) > ln(0.2) \implies max(n) > \dfrac{ln(0.2)}{ln(0.95)} > 31 \implies$

$max(n) = 31.$

Let's check that answer.

$1 - 0.95^{31} \approx 0.796 < 0.8,\ but\ 1 - 0.95^{32} \approx 0.806 > .8.$

Do you follow that?
Yes, I do understand your working step. Thank you mate.
palongze is offline  
Reply

  My Math Forum > High School Math Forum > Probability and Statistics

Tags
finding, probability



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
anyone have "secrets" for finding lowest terms with huge numbers. GIjoefan1976 Algebra 9 March 9th, 2016 08:59 PM
Can anyone help me finding this book "Evariste Galois "Toti" johnmath Math Books 6 January 27th, 2013 04:13 PM
A "simple" application of dirac delta "shift theorem"...help SedaKhold Calculus 0 February 13th, 2012 12:45 PM
"separate and integrate" or "Orangutang method" The Chaz Calculus 1 August 5th, 2011 10:03 PM
sample exeriment-need help finding "statistic" and "result" katie0127 Advanced Statistics 0 December 3rd, 2008 02:54 PM





Copyright © 2017 My Math Forum. All rights reserved.