Re: Formulas for speeding up repetitive rolls
As pointed out by others, the probability of success is defined on p.B171. That's all you need to know to do the math.
As to what math: To determine the number of successes achieved over a specific time period/specific number of rolls: multiply the number of rolls by the probability of success of your effective skills/attribute to determine how many rolls will be successful, and drop fractions. Example (same as provided by a previous post): 30 days of healing means 30 rolls. If your effective HT for healing is 11 (62.5%), that would give you (30 x 0.625 = 18.75, dropping fractions gives) 18 successes, so you would regain 18 HP over 30 days.To determine how many rolls/how much time is required to achieve a certain number of successes: divide the number of successful rolls required by the probability of success, then round up. Example: You need to recover 12 HP, thus make 12 daily healing rolls. Your effective HT for healing is 11 (62.5%), you would need to make (12 / 0.625 = 19.2, rounded up to) 20 healing rolls, or it takes 20 days to fully recover. 
Re: Formulas for speeding up repetitive rolls
Since the healing rate was stated as the example, here is the statistically average rates, with a bit of rounding to make nice numbers.
Effective HT for healing (after all modifiers) > rate of HP recovery HT 3 or less = 1 HP per 200 days (or 0.005 HP per day) HT 4 = 1 HP per 50 days (or 0.02 HP per day) HT 5 = 1 HP per 20 days (or 0.05 HP per day) HT 6 = 1 HP per 10 days (or 0.1 HP per day) HT 7 = 1 HP per 7 days (or 0.15 HP per day) HT 8 = 1 HP per 5 days (or 0.2 HP per day) HT 9 = 1 HP per 3 days (or 0.33 HP per day) HT 10 = 1 HP per 2 days (or 0.5 HP per day) HT 11 = 1 HP per 1.5 days (or 0.66 HP per day) HT 12 = 1 HP per 1.3 days (or 0.75 HP per day) HT 13 = 1 HP per 1.25 days (or 0.8 HP per day) HT 14 = 1 HP per 1.1 days (or 0.9 HP per day) HT 15 = 1 HP per 1.05 days (or 0.95 HP per day) HT 16 or more = 1 HP per 1.02 days (or 0.98 HP per day) An effective healing HT of 17 or higher is no better than 16 since a roll of 17 or more is an automatic failure. FYI, personally I'd treat an effective HT of 14 or more as 1 HP per day, as those fractions as close enough to not really matter. 
Re: Formulas for speeding up repetitive rolls
Ritual Path Magic is based on repetitive rolls, and there's a table somewhere (I think it's in one of the Monster Hunters supplements) that's designed to collapse a large series of repetitive rolls into one. Perhaps that could adapted for use with other kinds of repetitive rolls?

Re: Formulas for speeding up repetitive rolls
A variation on the idea of just using the % success/failure rate of your skill level is to first have a few dice rolls that can modify this; I suggested such a system here. You can speed things up and get roughly comparable results by instead using 3dF, as I noted later in the thread. It typically requires a calculator (or a player with the Lightning Calculator Advantage) to use, but I'm pretty certain most of us carry one of those around at all times anyway (as an app on our phones).
EDIT: Let's go with an injured PC laying in a TL 8 hospital for 30 days. The PC has HT 10, and the hospital staff is working with an effective Physician 12. Normally, this would require 30 rolls against 11 (HT 10, +1 for Physician 12+) for natural recovery and 30 rolls against 12 for assisted recovery. Under my suggestion, we can either roll these separately, for 3 rolls each (total 6 instead of 60), or even just apply the same adjustment to each (for 3 rolls instead of 60  or 30 if above we simply made a single roll and checked it against both HT and Physician). For this demonstration, I'll do both. For the natural recovery rolls, I'm getting 13, 12, and 16 (bad luck on those rolls), for a total of 9; that puts the character at an effective value of 2, so he fails to recover on his own at all (fortunately, Critical Failure doesn't do anything on natural recover rolls, so we don't need to look at that). For the assisted recovery, I'm getting 8, 8, and 11, for a total of +3; that puts things at 15, or 95.37% success rate  that's 28.61 successes. With our initial skill 12, MoF 5+ (a roll of 17 or 18) is needed for a Critical Failure, but we hit a 0% failure rate at just MoS 3, so we're safe. A Critical Success, meanwhile, requires MoS 8+ (roll of 4 or lower); that corresponds to 7 on our table, so 16.20% of our rolls  4.86  are Critical Successes, each of which restores 1 additional HP. So, our injured character recovers a total of 0 + 28.61 + 4.86 = 33.47 HP, which rounds down to 33 HP. Note using 3dF (which if you lack dF  like me  you can make do with setting 1,2 as +, 3,4 as 0, and 5,6 as ) makes things markedly less swingy. Even the worst roll above, 16 (which was 556 IIRC), would have only been a 3, letting the character heal for 7.779 additional HP, while the best, 8 (which was 224) would have been good for a +2. Rerolling with essentially 3dF, I'm getting +0 (245) and +2 (123). That puts the natural healing at 11, for 62.50% success  18.75 (natural healing doesn't care about criticals in either direction, so no need to check for that). Our assisted healing is at 14, for 90.74% success  27.22. For skill 12, you need MoF 5 for Critical Failure (which isn't an issue here, as we hit 0% failure at MoF 4), and MoS 8 for Critical Success; the latter corresponds to 6, so 9.26%  2.79  heal an additional 1 HP. Total healing is therefore 18.75 + 27.22 + 2.79 = 48.76, which rounds up to 49 HP. 
Re: Formulas for speeding up repetitive rolls
That is really far too complicated, especially since we only need to multiply probability by number of attempts. Anyway, no TL8 hospital will keep anyone in a bed after they get above (HP), as hospital beds are too valuable (and far too expensive in the USA). You would need to be able to pay out of pocket for daily care from a personal physician, or have such an individual as a good friend/relative (a really good use of Ally), to benefit from physician care past (HP).

Re: Formulas for speeding up repetitive rolls
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Re: Formulas for speeding up repetitive rolls
A TL9 automed would be an option, especially on the yacht of a wealthy traveler.

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I certainly agree they could and would release patients with positive hp, unless there is a condition unrelated to hp that require hospitalisation . But patient with negative hp ? That seem dubious, even for patient suffering from pure physical damage. 
Re: Formulas for speeding up repetitive rolls
If the patient is in no danger of dying from injury or infection, they will send them home (at least in the USA). Hospitals are dangerous environments, they have lots of sick people, and it is often safer (if slower) for patients to recover at home than at a hospital. They may have a nurse visit them once a day to change bandages, but that is only assuming that there is no one in the household to help the patient.

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