In the book "Bridge odds for practical players" , chapter 6 changing odds, there is an analysis on how the odds are changed during the play.
For those new to bridge, a suit contains 13 cards and it's important to know the chances of how it breaks between opponents (i.e. if you have 7 cards, the opponents have 6 and you want to know if it breaks 3-3, 4-2, 5-1 or 6-0).
When the North-South cards are AKQ53 - 42, it says that (as known) that the probability that the Jack will fall on the AKQ (i.e. the suit breaks 3-3 to the opponents) is ~42% and ~58% for the 4-2.
However, when the North-South cards are AKQ103 - 42, the following statement is given:
Again both defenders are following suit when you play A and K. Is it again a 42% chance that the suit will break 3-3? Not so. The difference in this time is that the jack is a significant card, in the sense that a defender would never play it unless forced to do so. In applying the deletion principle we have to rule out not only the 6-0 and 5-1 distributions, but also all the 4-2 that contain a doublenton Jack, i.e. 1/3 of these. The 3-3 divitions retain their full weight because a defender wouldn't make things easy for you by ropping the jack on the first or second round from Jxx. So we are left with odds for the 3-3- and the remaining 4-2 distributions in the ration of 35.52 to 32.2, or 11 to 10. The probability of the 3-3 break has risen to 52.4%"
Is the above reasoning mathematically sound? If yes, why one would consider a 50% finesse from the beginning when the holding is AKQ103?
Best
Diamond Life Master here. Yes, the reasoning is sound. The point is that it's a conditional probability problem where you know that no one is voluntarily playing the Jack.
You should not finesse from the beginning when your holding is $\text{AKQ10x}$ facing $\text{xx}$. Playing the suit from the top wins (a) whenever the suit divides $3$-$3$, and (b) whenever the suit divides unevenly and the Jack is in the short hand. Playing the suit from the top wins more than $50$% of the time.
In the similar situation where you have $\text{AKQ10}$ facing $\text{xxx}$, when you lead the third round of the suit toward the strong hand and your opponent follows small, at this point there are only $10$ unknown cards in that opponent's hand, while the opponent's partner still has $11$ unknown cards. That means the Jack is more likely to be in the partner's hand (the odds are $11:10$), so you should play for the drop.
I'll add, by the way, that in the first situation ($\text{AKQxx}$ facing $\text{xx}$), it's not correct to say that the Jack will fall less than $50$% of the time, as you write above. The Jack still will fall whenever the suit divides evenly and also whenever the Jack is in the short hand, and that still happens more than $50$% of the time. It may even fall more often, because if a defender holds $\text{J1098}$, that defender may play the Jack without being forced to.
The issue, though, is that your spots are so weak that even if the Jack falls, you still can't win five tricks if the suit splits $4$-$2$.