It's improbable, or not?

When people think about improbable events, often a challenge is made such as the one I read of involving Bertrand Russell (this may be apochryphal). Russell was said to have been riding in a car with someone, he noted the license plate on the car in front of them, and remarked, "Look at that license plate! What is the probability that we two would be riding on this road at this exact moment of time with these clouds overhead and this exact song playing on the radio and seeing this license plate in front of us? Astronomically low, therefore God must exist!" Or words to that effect.
 
Now such examples are at times presented as the probability argument of Intelligent Design (called often simply ID), where a structure is noted, its development deemed remotely improbable, and then the conclusion is made that a Designer designed it.
 
First, we may note that some event was going to happen when Russell and his friend were driving down the street, some license plate or other would likely appear in front of them, and some structure or other would reasonably evolve by the processes of evolution, but is some structure such as a flagellum virtually certain to evolve one way or the other? If so, then the design argument is done for, and this analogy is apt.
But since some flagellum was not certain to evolve, this analogy is then applying a situation where some event will occur, to a situation where one event may occur, and calling the two analagous, when in fact they are not.
 
To illustrate this, let's say we have 1 red ball with 9 blue balls in a box. Now what is the probability that we draw the red ball? 10%, certainly. What is the probability that we draw any ball? Well, 100%.
 
But is the ID argument saying that we drew some ball? No, it's checking to see how often we could draw a blue one. Now if we number the balls 1 to 10, then the probability of drawing any particular ball is 10%, and is this what the ID argument is doing? Again, no, for the set from which the choice is made is not the set of all possible biological structures, it is a specific set (and yet not just one exact flagellum, this sometimes is an objection). So if some set of outcomes is selected, which is a real biological structure which can be considered as an entity in itself, and that structure may or may not develop, then this is not the same as seeing that some structure developed, any structure, and then saying "How improbable to predict that!"

Now another source of misunderstanding comes when we consider whether an event that has occurred has probability 100% or maybe some other value. The important factor here is the reference point in time that we are considering, either before the event (this will be by imagination if the event has really happened) or after the event. To make this concrete, let's again pick one ball out of a set of 10 balls where one is blue and the others are red. I picked the blue ball, so what is the probability? Well, it's 10% with the frame of reference before the event, and 100% afterwards. But how can one event have two probabilities?
Well, let's view probabilities as predictions, that can sometimes help. Note by the way that a prediction of an event will be true the same number of times the event occurs, so the probability of a true prediction is the same as the probability of the event occurring. Now if I predict before the event that the ball will be blue, I will be right 10% of the time. If I predict a blue ball after the event, knowing the ball picked was the blue one, I will be right 100% of the time.
 
So the frame of reference in time indeed affects the probability. This also can be viewed as an example of conditional probability, that is, how much information do we start with?
 
If we know we picked the blue ball, that affects the probability, that gives us information that allows our prediction to be 100% right. The less we know, the less information that is given, the less accurate our prediction will be.
 
So if we are given the information that flagella evolved, isn't that like shooting an arrow and then drawing a target around it and crying "Bull's eye!"? What is known as the Texas sharpshooter fallacy. Is this what the ID argument is doing?
 
Well, again we must consider the frame of reference in time, for some naturalists do say the evolution of flagella is probable, now if it turns out their estimate is wrong, and the correct estimate has the probability quite small, does that turn it into a fallacy? Well, no.
 
And did they forget that flagella evolved, and they should have estimated not "probable," but 100% certain? Well, no.
 
Again, the probability of an event can be estimated with the reference point in time being before the event, or afterwards. Now if the point of reference is before the event, then we are asking to run the trial again, and see if we get the event of interest. Then run it again, and see if it happens.
 
The probability then is approximately the number of times the event occurred, divided by the number of times we tried to get it, and this gets more accurate, the more trials we do, given that our setup and the process isn't changing.
 
So this is not seeing the event occurred, and then claiming we knew it all along, and with such improbability! That would be drawing the target around the arrow. Instead, we are rewinding the tape, and (imagining) trying again and again, to see what would result.
 
If the probability of the event is then inordinately improbable, by natural processes, then we conclude some agent or process outside of nature was at work, if we feel confident we know all the relevant physical processes. Just as we would reject a claim that Farmer Bob's cow jumped the moon, we would say we know of the relevant physical processes, and don't expect a cow-launching jumping ability to be described in tomorrow's newspapers.

But if the probability is the number of times an event occurs over many trials, divided by the number of trials, how can we estimate that in the case of flagella? For we can't count successes, for we can't do all trials, nor can we enumerate all possible ways to try and generate flagella, to get the number of trials, both in fact are infinite.
 
So what are naturalists doing when they estimate that evolution of flagella is probable? They are examining scenarios, and estimating the probability of different steps, and by picking the scenario they deem most likely, coming up with an estimate of a lower bound of the probability, so the probability might be higher, but it's at least what they estimate, or approximately so.
 
The ID argument takes a similar tack, only the estimate is of steps that are seen to be needed in all scenarios, in forming the new structure, and this gives an upper bound on the probability, if the steps are relatively independent, especially, since then the probability of these steps can be multiplied to give an approximate upper bound on the probability.
 
Then if this upper bound is extremely small, we need not examine all paths, or all steps, we can conclude nature cannot reasonably be said to have produced this structure.