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<h1 id="1d4-hours-chance-of-waking-up-turn-by-turn">1d4 hours, chance of waking up turn by turn</h1>
<p>I was reading through Time Stop spell and it says you have 1d4+1 minutes
to act and I was daydreaming that you didn’t know when it ended;
that’s not the RAW because in it you kinda get a feel for when it’s
supposed to end, but it got me thinking to a similar issue we’ve been
talking about recently. When you’re unconscious for 1d4 hours, or when
you’re lost for 1d6 hours, or for how long it takes for hirelings to
come to the hamlet.</p>
<p>Those rules are fast and work well but I was curious just for
curiosity’s sake how to figure out a cumulative probability so you
could make a mechanic where you rolled every hour to see if you woke up.</p>
<p>I headed down a couple of wrong trails early on but what I ended up with
was simple.</p>
<p>Let’s say you have to wait/sleep/walk for X hours/days/weeks, where X
is some die size like 1d6. The good old example of sleeping for 1d4
hours after getting hurt is one of these.</p>
<p>First, you’ve got to sleep at least one hour. What are your chances of
waking up right after it? It’s ¼, or ¹⁄ₓ in the more general case. If
you fail that wake up roll and sleep more, the next chance is ⅓, or
¹⁄₍ₓ₋₁₎. If you fail that, and sleep more, it’s ½ chance of waking up,
or ¹⁄₍ₓ₋₂₎ in the general case. You keep going until you get to a ¹⁄₁
chance, in which case you obviously don’t roll.</p>
<p>To me it’s completely unintuitive that this should work, that this
should give the same probs. But it does work. I simplified it backwards
from a bigger equation with summed series etc, then double checked:</p>
<ul>
<li>Chance of waking up in the first of 1d4 hours: ¼, or 25%.</li>
<li>Chance of waking up in the second of 1d4 hours: ¾×⅓, or 25%.</li>
<li>Chance of waking up in the third of 1d4 hours: ¾×⅔×½, or 25%.</li>
<li>Chance of waking up in the fourth of 1d4 hours: ¾×⅔×½×1, or 25%.</li>
</ul>
<p>The same principle works for 1d6 hours, 1d12 days or whatever. It looks
bloody obvious to me now that I’ve spelled it out like this and I’m
probably gonna get a lot of “Well, duh…” but my first intuitions of
how a mechanic like this would look were quite far from working out and
it was third times the charm as I finally worked it out and then I
reordered it and got it in this order and was like “Ok, yeah, then
duh… now that I see <em>this</em>…” It was sort of a <a href="https://en.wikipedia.org/wiki/Monty_Hall_problem">Monte Hall
problem</a> for me until
everything clicked into place.</p>
<p><img src="https://idiomdrottning.org/cumulative-awakening.png" alt="A red cell and three empty cells, then one with two empty, then with one empty, then only the red cell alone" title="Cumulative Awakening" /></p>
<p>So in practice for sleeping 1d4 hours:</p>
<ol>
<li>Sleep through the first hour, then:</li>
<li>Roll 4 on 1d4 to wake up, otherwise sleep through the second hour
and:</li>
<li>Roll 3 on 1d3 to wake up, otherwise sleep through the third hour
and:</li>
<li>Roll 2 on 1d2 to wake up, otherwise sleep through the fourth hour
and then you wake up.</li>
</ol>
<p>For 1d4+1 minute, just go through an extra minute before you start.</p>
<p>For being lost for 1d6 hours:</p>
<ol>
<li>Wander around for the first hour, then:</li>
<li>Roll 6 on 1d6 to get a new chance to roll Survival, otherwise
wander around for another hour and:</li>
<li>Roll 5 on 1d5 to get a new chance to roll Survival, otherwise wander
around for another hour and:</li>
<li>Roll 4 on 1d4 to get a new chance to roll
Survival, otherwise wander around for another hour and:</li>
<li>Roll 3 on 1d3 to get a new chance to roll Survival,
otherwise wander around for another hour and:</li>
<li>Roll 2 on 1d2 to get a new chance to roll Survival,
otherwise wander around for another hour and then you’e get a new
chance to roll Survival.</li>
</ol>
<h2 id="how-to-deal-with-skipped-hours">How to deal with skipped hours?</h2>
<p><em>(This section was added 2019-05-22)</em></p>
<p>So I’ve been thinking about this and have come up with a new thing.</p>
<p>Skipped hours come up often; say you failed the 1/4 check after the
first of four hours and then it’s two hours later [three hours
total]; are you awake? I.e. you need to succeed on <strong>either</strong> 1/3 or
1/2. How to do that with one roll? It’s 2/3.</p>
<p>Use the denominator for the first check but instead of the numerator
being one, put in the amount of checks. So let’s say it’s been three
hours and the party needs to know if their Sleepy Beauty has awakened
yet. You could do three checks 1/4, 1/3, 1/2, or you could do one 3/4
check which is mathematically the same thing.</p>
<h2 id="multiple-dice">Multiple dice</h2>
<p>Sometimes a spell affects you for 3d4 rounds or whatever. That means
going through three full sequences of 1d4, 1d3, 1d2, 1d1 etc (stopping
each sequence when you max out the roll).</p>
<h2 id="when-to-use-this">When to use this</h2>
<p>This was one of the first house rules we added but it’s still among
the ones we like best. Obviously, we only use it when we’re “zoomed
in” enough (time-wise) for this fine granularity to matter. If we’re
taking an hour long short rest, we don’t need to go “round by round”
when every round is 6 seconds. We just roll the dice normally. But we
still end up finding uses for this rule almost every session, usually
multiple times.</p>
<p>And equally obviously, if you use hidden rolls you don’t need this.
That’s the gordian solution. But we prefer using open rolls for this,
especially when it matters because of encounter checks and other
dangers at the relevant time scale.</p>
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