Cut the knot: learn to enjoy mathematics
A math books store at a unique math study site. Learn to enjoy mathematics.
Google
Web CTK
Best sites for teachers
Sites for teachers
Sites for parents
Terms of use
Awards

Interactive Activities
CTK Exchange
CTK Insights - a blog

Games & Puzzles
What Is What
Arithmetic/Algebra
Geometry
Probability
Outline Mathematics
Make an Identity
Book Reviews
Eye Opener
Analog Gadgets
Inventor's Paradox
Did you know?...
Proofs
Math as Language
Things Impossible
Visual Illusions
My Logo
Math Poll
Cut The Knot!
MSET99 Talk
Other Math sites
Front Page
Movie shortcuts
Personal info
Reciprocal links
Privacy Policy

Guest book
News sites

Recommend this site

Best sites for teachers
Sites for teachers
Sites for parents
Education & Parenting

Manifesto: what CTK is about Search CTK Buying a book is a commitment to learning Table of content Things you can find on CTK Chronology of updates Email to Cut The Knot Recommend this page

Balls of Two Colors

An urn contains w white balls and b black balls (w > 0 and b > 0). The balls are thoroughly mixed and two are drawn, one after the other, without replacement. Let Wi and Bi denote the respective outcomes 'white on the ith draw' and 'black on the ith draw,' for i = 1, 2.

Prove that P(W2) = P(W1) = w/(w + b). (Which clearly implies a similar identity for B2 and B1.)

Furthermore, P(Wi) = w/(w + b), for any i not exceeding the total number of balls w + b.

Solution

Copyright © 1996-2008 Alexander Bogomolny

 

 

 

 

 

 

 

 

 

 

 

 

We must remember the formula for the total probability:

P(W2) = P(W2|W1)·P(W1) + P(W2|B1)·P(B1),

from which

P(W2) = (w - 1)/(b + w - 1)·w/(b + w) + w/(b + w - 1)·b/(b + w) =
w/(b + w - 1)(b + w)·(w - 1 + b),

which is simplified to w/(w + b).

References

  1. Ruma Falk, Understanding Probability and Statistics, A K Peters, 1993

Copyright © 1996-2008 Alexander Bogomolny

 

 

 

 

 

 

 

 

 

 

 

 

28733083Page copy protected against web site content infringement by Copyscape


Search:
Keywords:

Latest on CTK Exchange
Math
Posted by Laura
2 messages
06:56 AM, Apr-15-08
Divisibility rules - Jargon buste ...
Posted by Carolyn
2 messages
08:35 AM, Apr-04-08
drawing puzzle
Posted by martin gran
31 messages
06:53 PM, May-09-08
conway's game of life
Posted by frequency
0 messages
11:52 PM, May-12-08
Mistake on the page (an aside, Be ...
Posted by Max
4 messages
10:28 AM, Feb-28-08
Deriving functions based on diffe ...
Posted by ke_45
1 messages
12:47 PM, May-10-08
A typo in
Posted by alexwajn
1 messages
11:36 PM, Apr-19-08