63% start drowning, 54% drown
For this answer I am assuming the investigator is not insane, is in his 20s or 30s (no changes to rolled CON) and the swimming is of regular difficulty. Interestingly, the rules make no notice of clothing affecting the difficulty level of swimming. We assume that an investigator who "has fallen into turbulent water" had no time to undress, and regular difficulty applies to swimming in clothes. We also assume that no help is forthcoming, as per your conditions.
For the record, I feel that your concern about the probabilities is misplaced, as the game's intent clearly and strongly is that the keeper can and should keep the character alive even if they can let them drown according to the dice rolls, so the probabilities do not really matter. But you are asking explictly about the mathematical probabilities, so this is what I'll give.
Assume that constitution and all other ability scores are random (3d6 × 5).
Probabilty to make CON roll: 52.5%
Probablity to fail CON roll: 47.5%
Event chain probabilities with outcome
All swimmers that failed their CON or pushed roll started drowning, which is not the same as drowned, in case safety can be found by successfuly pushing after a failed CON roll.
Assume that 80% of characters have a 20 in swim
- Swim success: 20% survive
- Swim fail, CON success, pushed success: 80% * 52.5% * 20% = 8.4% survive
- Swim fail, CON fail, pushed success 80% * 47.5% * 20% = 7.6% survive
- Swim fail, CON success, pushed fail 80% * 52.5% * 80% = 33.6% drown
- Swim fail, CON fail, pushed fail 80% * 47.5% * 80% = 30.4% drown
Total sum of survive: 36%
Total sum of drown: 64%
Total sum of started drowning: 71.6%
Assume that 20% of characters have a 60 in swim.
- Swim success: 60% survive
- Swim fail, CON success, pushed success: 40% * 52.5% * 60% = 12.6% survive
- Swim fail, CON fail, pushed success 40% * 47.5% * 60% = 11.4% survive
- Swim fail, CON success, pushed fail 40% * 52.5% * 40% = 8.4% drown
- Swim fail, CON fail, pushed fail 40% * 47.5% * 40% = 7.6% drown
Total sum of survive: 84%
Total sum of drown: 16%
Total sum of started drowning: 27.4%
Assume that 80% of characters have a 20 in swim, and 20% of characters have a 60 in swim.
Results weighted by 80% for bad swimmers, 20% for good swimmers.
Total sum of survive: 80% * 36% + 20% * 84% = 45.6%
Total sum of drown: 80% * 64% + 20% * 16% = 54.4%
Total sum of started drowning: 80% * 71.6% + 20% * 27.4% = 62.76%
Swimming to safety (drowning):
An investigator has fallen into turbulent water and must
swim to safety or drown. The Keeper calls for a swim roll
with the goal “swim to safety.”
Making the roll means the investigator swims to safety and lives.
If the player loses, then no progress is made and water may be inhaled; the player must make a CON roll or his or her investigator will suffer 1D6 damage per round
This situation is already drowning: the character has swallowed water and will take 1D6 damage1 per round. You always roll CON if you fail the initial swim roll. The procedure continues:
The situation demands a pushed roll—the only alternative
is that the investigator gives up and drowns. If the player
misses the pushed swim roll, the investigator is battered and
half-drowned, taking 1D6 damage per round.
You always roll the pushed roll, wether you succeeded on the CON roll or not. Succeeding on the pushed roll means the investigator swims to safety and lives, because as per the rules they achieve their goal as if it would have been for the original roll. This is a bit weird as it does not matter if they already had started drowning, so it does not matter that they take 1D6 per round2.
The Keeper must then make an important decision: either the investigator’s
life is put on the line or the investigator is washed up later elsewhere. If the Keeper chooses the former, then the investigator will continue to lose hit points each round until saved by another investigator or non-player character. Alternatively, if no one else is around, the Keeper could waive the drowning damage and instead have the unsuccessful pushed roll mean that the investigator has washed up on some foreign shore, bereft of all possessions and in a bad way.
If the investigator fails their pushed swim roll, they will be drowning no matter what, whether or not they made their CON roll, as they now take damage as per the drowning rules. (We assume this is your choice as keeper, otherwise the whole exercise here has no point), and no further progress is to be made. No option of continuing swimming to safety is given, only the option of being saved by others. Either the character is saved by someone else, or he will die in the third round if they failed their CON roll, or two rounds later, if they made it. As we assume nobody is saving them, they drown either way.
Opposing Skill/Difficulty Level
The Regular difficulty level (requiring a roll of equal to or below the skill value) is the default roll.
Pushing the Roll
Pushing a skill roll provides the player with a second and
final attempt to achieve a goal. A pushed roll is only allowed
if it can be justified, and it is up to the player to do this.
Pushed Roll: Success The player’s goal is achieved as it would have been for the original roll. None of the consequences of failure happen.
Other forms of Damage
Asphyxiation and Drowning: a CON roll should be made each round; once a CON roll is failed, damage is sustained each round thereafter until death or until the victim is able to breathe. If the character is in a state of physical exertion, a Hard success is required on the CON roll.
1 One question is if drowning damage follows the rules for Combat Damage. If yes, on a roll of 5 to 6 (damage equal to more than half maximum hit points) then a separate CON roll would be called for; if failed the character would fall unconscious, could not swim and would drown outright. However, the rules text for Major Wound: Effects is this: A Major Wound results when an attack delivers an amount of damage equal to or greater than half of the character’s maximum hit points in a single attack , and as drowning is not an attack, I think this rule can be ignored here.
2 If you instead assume they take damage every round while swimming to safety, then it will matter if they made their CON roll. If they did not, hit points influcence how long they survive swimming and drowning. According to the combat rules, taking an action that requires a roll takes up a round, so the pushed swim roll takes up the second round. With average stats also for size, an ivestigator would have 8 hit points, and on average will die in the third round, unless someone intervenes or they reach saftey before then. Without a probability of that being the case, we cannot asses the overall probablity. But it would at best be as good as the numbers above.