Writer Stephen Dawson, a long-time participant at this blog, wrote and asked the following:
In his The Moon is a Harsh Mistress, Robert Heinlein had his intelligent computer calculate the odds of a successful revolution for independence of the moon, which turned out to be a low number. The conspirators embarked on their quest and after a while, although successful to that point, were disheartened to find that the odds of ultimate success had fallen, and continued to fall. From memory, Mike, the computer, explained that there were various paths to success, but that selection of any one thereby eliminated alternatives, reducing the odds. As they proceeded, the odds continued to reduce.
Eventually, of course, they neared their goal sufficiently for the odds to begin increasing.
Completely putting aside the question of whether such a calculation could reasonably be made in the first place, this idea that making a choice from one of several mutually exclusive paths will lower the odds of success has always troubled me. Do you think that, wonderful as he was, Heinlein was wrong on this?
Heinlein was right. Here’s why.
All statements of logic, including probability statements about the success of revolution, are conditional on certain evidence. Thus we cannot say, “The probability of showing a 5 spot on a die roll” is 1 in 6 without adding the evidence, “Given that we have a six-sided die, only one spot of which will show when tossed, and only one side of which is labeled 5.”
Keep that evidence in mind and consider this scenario, which we can call the revolution game: you will roll a die twice; if you roll at least a 5 on the first, you are allowed a second roll (else not), and if this second roll is a 6 you win.
What are the chances your revolution succeeds? Well, rolling at least a 5, given our evidence and game rules, is 2/6, and then rolling a 6 the second time is 1/6, so together the chance is 2/36 = 1/18. Not so high.
But once you “go down the path” of rolling a 5 or 6 on the first roll, then your chance of winning soars by three times to 6/36, or 1/6. Because, of course, your evidence that you use to compute the probability has changed based on what has happened.
This is no different than considering a situation where a mythical traveler is presented with several doorways, all but one of which open to his destination. If the traveler has more information than just the number of doorways, then he can increase his chance of arriving by applying that information. Perhaps, for instance, he has learned that the real doorway will be “hand crafted” and that the false will be machine made. That additional information changes the probability of succeeding, though perhaps not in a perfectly quantifiable way.
(One delusion to which we Moderns succumb is that quantification is always possible, usually to arbitrary precision.)
Or think of it this way. Suppose you want to operate on a fellow with an operable disease. Given our knowledge of surgery, what are the chances you save him? First consider that you haven’t gone to medical school. The odds of success are low. But after slogging through for four years of school, seven or eight years of residency and a fellowship—all choices you made—then the odds of a successful outcome increase dramatically. Because the evidence used to the compute that probability has changed.
Update I read the problem backwards; it should be a decreasing, not increasing, probability along a path, but one which turns into increasing probability. This is still possible.
I don’t want to include any complex drawing. But imagine in front of you are several paths. You take one and then learn, via Mike and given the information available, your probability of success has decreased; it would have increased or not decreased as much had you taken another path. Perhaps on this path, no matter what, you to win the lottery at least once to continue.
You take another path (already having progressed down the first) and learn that you now have to win the lottery twice. And so on.
If I have time later, I’ll include a numerical example.