DaWolf85
Non-stop Poster
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Hey guys! It's been a long 10 years since anyone has made a post about the probability of a noble train succeeding.
In that time, items have come out that boost the chances of you taking a village, and new paladin skills have been introduced that do the same thing. So I figured I'd update the math to fit our new item-and-paladin-based reality. In addition, I figured I'd calculate the probabilities of you nobling the village from yourself with a given train (over-nobling). I've added a spoiler here so you can see my methodology, and replicate the results yourself, or offer feedback on my methods:
Before we begin I would also like to go over what buffs are possible.
Items can give +2 for the smaller booster, or +5 for the larger one. Persuasion paladins can give +2, +4, +6, or +10 depending on level. That means our possible buffs are:
I will be calculating probabilities for all of these scenarios.
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So let's get on with it! Here are the numbers.
A 3-train without any buffs has a 1.37% chance of taking the village, and cannot over-noble itself.
A 4-train without any buffs has an 86.70% chance of taking the village, and a 0.94% chance of over-nobling once. It cannot over-noble twice.
A 5-train without any buffs has a 100% chance of taking the village, a 59.60% chance of over-nobling at least once, and a 0.65% chance of over-nobling twice.
A 3-train with a +2 buff has an 8.89% chance of taking the village, and cannot over-noble itself.
A 4-train with a +2 buff has a 97.92% chance of taking the village, and a 6.11% chance of over-nobling once. It cannot over-noble twice.
A 5-train with a +2 buff has a 100% chance of taking the village, a 67.32% chance of over-nobling at least once, and a 4.20% chance of over-nobling twice.
A 3-train with a +4 buff has a 27.54% chance of taking the village, and cannot over-noble itself.
A 4-train with a +4 buff has a 99.95% chance of taking the village, and an 18.93% chance of over-nobling once. It cannot over-noble twice.
A 5-train with a +4 buff has a 100% chance of taking the village, a 68.72% chance of over-nobling at least once, and a 13.02% chance of over-nobling twice.
A 3-train with a +5 buff has a 40.67% chance of taking the village, and cannot over-noble itself.
A 4-train with a +5 buff has a 100% chance of taking the village, and a 27.96% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +6 buff has a 54.69% chance of taking the village, and cannot over-noble itself.
A 4-train with a +6 buff has a 100% chance of taking the village, and a 37.60% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +7 buff has a 68.26% chance of taking the village, and cannot over-noble itself.
A 4-train with a +7 buff has a 100% chance of taking the village, and a 46.93% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +8 buff has an 80.08% chance of taking the village, and cannot over-noble itself.
A 4-train with a +8 buff has a 100% chance of taking the village, and a 55.01% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +9 buff has an 88.89% chance of taking the village, and cannot over-noble itself.
A 4-train with a +9 buff has a 100% chance of taking the village, and a 61.11% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +10 buff has a 94.63% chance of taking the village, and cannot over-noble itself.
A 4-train with a +10 buff has a 100% chance of taking the village, and a 65.06% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +11 buff has a 97.95% chance of taking the village, and cannot over-noble itself.
A 4-train with a +11 buff has a 100% chance of taking the village, and a 67.34% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +12 buff has a 99.51% chance of taking the village, and cannot over-noble itself.
A 4-train with a +12 buff has a 100% chance of taking the village, and a 68.41% chance of over-nobling once. It cannot over-noble twice.
A 2-train with a +15 buff has a 0.39% chance of taking the village, and cannot over-noble itself.
A 3-train with a +15 buff has a 100% chance of taking the village, and a 0.27% chance of over-nobling once. It cannot over-noble twice.
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Those are my calculations. There was a lot of copy-pasting involved, and there is some chance I've made errors somewhere. Please let me know if you spot anything.
In that time, items have come out that boost the chances of you taking a village, and new paladin skills have been introduced that do the same thing. So I figured I'd update the math to fit our new item-and-paladin-based reality. In addition, I figured I'd calculate the probabilities of you nobling the village from yourself with a given train (over-nobling). I've added a spoiler here so you can see my methodology, and replicate the results yourself, or offer feedback on my methods:
I used anydice.com for this experiment. This is a site where you can set up dice rolls and it will give you the probabilities of each possible result. With some work, this can be used to simulate noble trains.
Here is the function that I used:
And here is what that function looks like being used to simulate a standard 4-train:
You can use this function yourself to recreate edge cases such as nobles from two different villages, one with a persuasion paladin and one without; as simulating that is beyond the scope of this post. For ease of pasting, here is a link to the code already inputted for a standard 4-train.
Over-nobling probabilities are calculated by multiplying the probability of a train of the necessary length capturing the village (for a 4-train over-nobling once, this would be a 3-train), by the probability of a single noble dropping at least 25 loyalty (68.75%) as many times as I am calculating the over-nobles (so for a 4-train over-nobling once, I would take the probability of a 3-train capturing the village, and multiply it by 68.75% only once). For another example, with a 5-train over-nobling twice, I would take the probability of a 3-train capturing the village, and multiply it by 68.75% twice.
I have left out train sizes that are redundant - if a size of train for a given buff is not listed, it either is too small to take the village on its own, or it is larger than the smallest train with a 100% chance to cap.
Here is the function that I used:
function: noble plus A {
result: 1d16+19+A
}And here is what that function looks like being used to simulate a standard 4-train:
output [noble plus 0]+[noble plus 0]+[noble plus 0]+[noble plus 0]You can use this function yourself to recreate edge cases such as nobles from two different villages, one with a persuasion paladin and one without; as simulating that is beyond the scope of this post. For ease of pasting, here is a link to the code already inputted for a standard 4-train.
Over-nobling probabilities are calculated by multiplying the probability of a train of the necessary length capturing the village (for a 4-train over-nobling once, this would be a 3-train), by the probability of a single noble dropping at least 25 loyalty (68.75%) as many times as I am calculating the over-nobles (so for a 4-train over-nobling once, I would take the probability of a 3-train capturing the village, and multiply it by 68.75% only once). For another example, with a 5-train over-nobling twice, I would take the probability of a 3-train capturing the village, and multiply it by 68.75% twice.
I have left out train sizes that are redundant - if a size of train for a given buff is not listed, it either is too small to take the village on its own, or it is larger than the smallest train with a 100% chance to cap.
Before we begin I would also like to go over what buffs are possible.
Items can give +2 for the smaller booster, or +5 for the larger one. Persuasion paladins can give +2, +4, +6, or +10 depending on level. That means our possible buffs are:
+0 (default)
+2 (small booster or level 1 persuasion)
+4 (small booster and level 1 persuasion; or level 2 persuasion)
+5 (large booster)
+6 (small booster and level 2 persuasion; or level 3 persuasion)
+7 (large booster and level 1 persuasion)
+8 (small booster and level 3 persuasion)
+9 (large booster and level 2 persuasion)
+10 (level 4 persuasion)
+11 (large booster and level 3 persuasion)
+12 (small booster and level 4 persuasion)
+15 (large booster and level 4 persuasion)
+2 (small booster or level 1 persuasion)
+4 (small booster and level 1 persuasion; or level 2 persuasion)
+5 (large booster)
+6 (small booster and level 2 persuasion; or level 3 persuasion)
+7 (large booster and level 1 persuasion)
+8 (small booster and level 3 persuasion)
+9 (large booster and level 2 persuasion)
+10 (level 4 persuasion)
+11 (large booster and level 3 persuasion)
+12 (small booster and level 4 persuasion)
+15 (large booster and level 4 persuasion)
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So let's get on with it! Here are the numbers.
A 3-train without any buffs has a 1.37% chance of taking the village, and cannot over-noble itself.
A 4-train without any buffs has an 86.70% chance of taking the village, and a 0.94% chance of over-nobling once. It cannot over-noble twice.
A 5-train without any buffs has a 100% chance of taking the village, a 59.60% chance of over-nobling at least once, and a 0.65% chance of over-nobling twice.
A 3-train with a +2 buff has an 8.89% chance of taking the village, and cannot over-noble itself.
A 4-train with a +2 buff has a 97.92% chance of taking the village, and a 6.11% chance of over-nobling once. It cannot over-noble twice.
A 5-train with a +2 buff has a 100% chance of taking the village, a 67.32% chance of over-nobling at least once, and a 4.20% chance of over-nobling twice.
A 3-train with a +4 buff has a 27.54% chance of taking the village, and cannot over-noble itself.
A 4-train with a +4 buff has a 99.95% chance of taking the village, and an 18.93% chance of over-nobling once. It cannot over-noble twice.
A 5-train with a +4 buff has a 100% chance of taking the village, a 68.72% chance of over-nobling at least once, and a 13.02% chance of over-nobling twice.
A 3-train with a +5 buff has a 40.67% chance of taking the village, and cannot over-noble itself.
A 4-train with a +5 buff has a 100% chance of taking the village, and a 27.96% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +6 buff has a 54.69% chance of taking the village, and cannot over-noble itself.
A 4-train with a +6 buff has a 100% chance of taking the village, and a 37.60% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +7 buff has a 68.26% chance of taking the village, and cannot over-noble itself.
A 4-train with a +7 buff has a 100% chance of taking the village, and a 46.93% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +8 buff has an 80.08% chance of taking the village, and cannot over-noble itself.
A 4-train with a +8 buff has a 100% chance of taking the village, and a 55.01% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +9 buff has an 88.89% chance of taking the village, and cannot over-noble itself.
A 4-train with a +9 buff has a 100% chance of taking the village, and a 61.11% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +10 buff has a 94.63% chance of taking the village, and cannot over-noble itself.
A 4-train with a +10 buff has a 100% chance of taking the village, and a 65.06% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +11 buff has a 97.95% chance of taking the village, and cannot over-noble itself.
A 4-train with a +11 buff has a 100% chance of taking the village, and a 67.34% chance of over-nobling once. It cannot over-noble twice.
A 3-train with a +12 buff has a 99.51% chance of taking the village, and cannot over-noble itself.
A 4-train with a +12 buff has a 100% chance of taking the village, and a 68.41% chance of over-nobling once. It cannot over-noble twice.
A 2-train with a +15 buff has a 0.39% chance of taking the village, and cannot over-noble itself.
A 3-train with a +15 buff has a 100% chance of taking the village, and a 0.27% chance of over-nobling once. It cannot over-noble twice.
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Those are my calculations. There was a lot of copy-pasting involved, and there is some chance I've made errors somewhere. Please let me know if you spot anything.
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