nmillar: There are 14 players in the game. Normally a player cannot use their ability on themselves, so there are 13 possible targets (my own role and a doctor role would be the exception - I can jail myself and a doctor could protect themselves).
As you quite rightly point out (and assuming random selection):
I have a 1 in 14 chance of jailing the person who is selected by the mafia to carry out the night kill.
There is a 1 in 11 chance of the person carrying out the night kill to target the person who has been jailed.
However, the laws of statistics state that for these 2 events to happen simultaneously, then you would multiply the fractions. In this case, giving a 1 in 154 chance (this differs from my previous number as I forgot I could jail keep myself).
I think that if you were honest, you wouldn't be requiring a maths lesson, and would notice, by yourself, the flaw in your rhetorics. You would realise spontaneously that, if, say, a doctor selects a target, he has 1/numbers_of_other_players chances to select the mafia's victim. If there is a doctor protecting someone, the mafia has 1/numbers_of_townies chances to target the protected person. Both probabilities o not combine.
If you roll a dice, you have 1/6 chances of matching a predefined number. Even if this predefined number was selected by a dice roll.
If you roll two dice, you have 1/6 chances of getting the same roll.
1-1
2-2
3-3
4-4
5-5
6-6
versus six times a variation on five remaining combination :
1-2
1-3
1-4
1-5
1-6
2-1
2-3
2-4
2-5
2-6
etc.
In other words, no, a doctor wouldn't have one-out-of-150 chances of protecting the victim every time. It would make the role virtually useless.
The same goes with jailing. You selected one target, its chances to be the target (~1/total) is the same as his chances of being the killer (~1/total). If you need to bring it down to details, it's the sum of victims he could have chosen, versus the chances of him being chosen as a victim multiplied by the possibilities of different people making the choice.
Or you can phrase it differently. You picked a number. There was (probably) going to be one killer and one victim. There was one chance out of total-possible-jail-targets for him to be the victim, and there was one chance out of total-possible-jail-targets for him to be the assassin. It's the same.
It's not a matter of choosing a calculation perspective. There are maths, and there are manipulations. You're not just "choosing the method (aka results) you want", when yu calculate probabilities. Your teacher won't take it for an excuse for grading your test if your answer is false.