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<h1>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 <i>this</i>…” 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="https://idiomdrottning.org/cumulative-awakening.png" /></p>
<p>So in practice for sleeping 1d4 hours:</p>
<ol>
<li>Sleep through the first hour, then:</li>
<li>Roll 10+ on 1d12 to wake up, otherwise sleep through the second hour and:</li>
<li>Roll 9+ on 1d12 to wake up, otherwise sleep through the third hour and:</li>
<li>Roll 7+ on 1d12 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 11+ on 1d12 to get a new chance to roll Survival, otherwise wander around for another hour and:</li>
<li>Roll 11+ on 1d12 but rerolling ones and twos until you get a non 1 or 2 result) to get a new chance to roll Survival, otherwise wander around for another hour and:</li>
<li>Roll 10+ on 1d12 (no rerolls this time) to get a new chance to roll Survival, otherwise wander around for another hour and:</li>
<li>Roll 9+ on 1d12 (no rerolls) to get a new chance to roll Survival, otherwise wander around for another hour and:</li>
<li>Roll 7+ on 1d12 (etc) 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>
<p>(It’s pretty much 1d6, 1d5, 1d4, 1d3, 1d2 and then a sure thing.)</p>
<p>For a d20 equivalent, it’s 11+ (2 hours), 14+ (3 hours), 16+ (4 hours), 17+ (5 hours), 18+ (6–8 hours), 19+ (9–13 hours), 20+ (14 or more hours).</p>
<h2>How to deal with skipped hours?</h2>
<p><i>(This section was added 2019-05-22)</i></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 <b>either</b> 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>
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