Problem #70

For more information on this collection, see Pillow-Problems by Charles L. Dodgson (Lewis Carroll). Note that Carroll most likely intended this problem, in the subject area of what he describes as "transcendental probabilities", to be a joke, not to be taken seriously.

For more information on this collection, see Pillow-Problems by Charles L. Dodgson (Lewis Carroll). Note that Carroll most likely intended this problem, in the subject area of what he describes as "transcendental probabilities", to be a joke, not to be taken seriously.

72.

A bag contains 2 counters, as to which nothing is known except that each is either black or white. Ascertain their colours without taking them out of the bag.

[8/9/87

72.

One is black, and the other white.

72.

We know that, if a bag contained 3 counters, 2 being black and one white, the chance of drawing a black one would be

; and that any (2)/ (3) *other* state of things would *not* give this chance.

Now the chances, that the given bag contains (α) `BB, (β) BW, (γ) WW, are respectively ¼, ½, ¼.
`

Add a black counter.

Then the chances, that it contains (α) `BBB`, (β) `BWB`, (γ) `WWB`, are, as before, ¼, ½, ¼.

Hence the chance, of now drawing the black one,

= ¼ · 1 + ½ ·

+ ¼ ·

=

.

(2)/ |

(3) |

(1)/ |

(3) |

(2)/ |

(3) |

Hence the bag now contains `BBW` (since any *other* state of things would *not* give this chance).

hence, before the black counter was added, it contained `BW`, i.e. one black counter and one white.

Q.E.F.

The end.