There are two bags, one containing a counter, known to be either white or black; the other containing 1 white and 2 black. A white is put into the first, the bag shaken, and a counter drawn out, which proves to be white. Which course will now give the best chance of drawing a white—to draw from one of the two bags without knowing which it is, or to empty one bag into the other and then draw?
The first course gives chance = ½; the second,
Hence the first is best.
The ‘a priori’ chances of possible states of first bag are ‘W, ½ B, ½’. Hence chances, after putting W in, are ‘WW, ½ WB, ½’. The3 chances, which these give to the ‘observed event’, are 1, ½. Hence chances of possible states ‘W, B’, after the event, are proportional to 1, ½ i.e. to 2, 1; i.e. their actual values are , .
Now, in first course, chance of drawing W is ½ · + ½ · ; i.e. ½.
And, in second course, chances of possible states ‘WWBB, WBBB’ are , : hence chance of drawing W is · ½ + · ¼; i.e. .
Hence first course gives best chance.