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What do you think of the following 'proof' by Lewis Carroll that an urn cannot contain two balls of the same colour? Suppose that the urn contains two balls, each of which is either black or white; thus, in the obvious notation, P(BB) = P(BW) = P(WB) = P(WW) = 1. We add a black ball, so that P(BBB) = P(BBW) = P(BWB) = P(BWW) = 1. Next we pick a ball at random; the chance that the ball is black is (using conditional probabilities) 1 · 4 + 3 · 4 + 3 · 4 + 3 · 4 = 3. However, if . there is probability that a ball, chosen randomly from three, is black, then there must be two black and one white, which is to say that originally there was one black and one white ball in the urn.

What do you think of the following 'proof' by Lewis Carroll that an urn cannot contain two balls of the same colour? Suppose that the urn contains two balls, each of which is either black or white; thus, in the obvious notation, P(BB) = P(BW) = P(WB) = P(WW) = 1. We add a black ball, so that P(BBB) = P(BBW) = P(BWB) = P(BWW) = 1. Next we pick a ball at random; the chance that the ball is black is (using conditional probabilities) 1 · 4 + 3 · 4 + 3 · 4 + 3 · 4 = 3. However, if . there is probability that a ball, chosen randomly from three, is black, then there must be two black and one white, which is to say that originally there was one black and one white ball in the urn.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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What do you think of the following 'proof' by Lewis Carroll that an urn cannot contain two balls of the same colour? Suppose that the urn contains two balls, each of which is either black or white; thus, in the obvious notation, P(BB): = P(BW) = P(WB) = P(WW) = 1. We add a black ball, so that P(BBB) = P(BBW) = P(BWB) = P(BWW) = 1. Next we pick a ball at random; the chance that the ball is black is (using conditional probabilities) 1.4 +3·+· 1+1=3. However, if there is probability that a ball, chosen randomly from three, is black, then there must be two black and one white, which is to say that originally there was one black and one white ball in the urn.
Transcribed Image Text:What do you think of the following 'proof' by Lewis Carroll that an urn cannot contain two balls of the same colour? Suppose that the urn contains two balls, each of which is either black or white; thus, in the obvious notation, P(BB): = P(BW) = P(WB) = P(WW) = 1. We add a black ball, so that P(BBB) = P(BBW) = P(BWB) = P(BWW) = 1. Next we pick a ball at random; the chance that the ball is black is (using conditional probabilities) 1.4 +3·+· 1+1=3. However, if there is probability that a ball, chosen randomly from three, is black, then there must be two black and one white, which is to say that originally there was one black and one white ball in the urn.
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