Conditional probability

The assumptions you make without noticing can change the answers you get.

Last week we described an essay on probability (that ventured into other areas as well) by the Marquis de Laplace.  As an example of a problem he posed and solved in the essay, we presented this version:

“To proceed on your Quest, you are given a leather pouch with two amulets.  An amulet may be either silver or gold.  You may not look into the pouch, but to pass a Gate you must draw one at random, replacing it afterward.  The amulet you draw for the first Gate is silver.  The one for the second Gate is also silver.  What is the probability that the one you draw for the third Gate is again silver?”

This is modified from Laplace’s original, which involved black and white balls and urns and (we thought) needed a bit of spicing up.  As an application of conditional probability our tutor thought it interesting enough to present as a challenge problem on his cubicle whiteboard, with results we’ll get to in a moment.  One of our correspondents turned it into a Dungeons & Dragons situation, casting a spell to change the amulet’s metal.  Well, D&D has many interesting applications of probability, but for the moment we’ll rule out transmutation.

Laplace proceeded along these lines: before anything is drawn there are three possibilities, a pouch with two silver, one of each, or two golds.  The last is ruled out by the first draw.  The first is a more probable source of a double silver by four to one.  Some multiplication gives a probability of nine-tenths that the third draw will be silver.  Note an analogy with science: given data (two silvers), we work out the probability of theories (type of pouch).

But some of our tutor’s colleagues approached the problem in a different way.  Suppose the pouch is constructed by choosing the first amulet, with a fifty-fifty chance of each metal; then the second.  There is only one way of getting two silvers, but two ways of getting a mixed bag, so before we draw anything the probability of the latter is higher by two-to-one.  Working through the conditionals we get a chance of five-sixths that the third amulet will be silver.  The difference is not great, but it is significant.  One tutor even worked out the consequences of different assumptions about the chance of gold versus silver.  The result was one could get a third-silver probability anywhere from just above 50% to just short of certainty, depending on one’s assumptions.

So the answers we get depend significantly on the assumptions we make before starting our calculation.  Put another way, how we ask a question in probability can have a big effect on the answer.  And unstated or unnoticed assumptions can be pernicious.