Exploration Meçhanics

[TheYell]

I have a kingdom map with a 10*10 grid. Each square represents 2500 sq miles. Each square has 1-8 tribes, each being about 1000 people.

Considering it will take about four days to cross a square, what should the odds be of meeting a tribe each day?

[aimlessPolymath]

This partly comes down to visibility, but really it comes down to the degree to which geography funnels people into similar travel routes.
Each square is 50x50 miles.

Any given group has a visibility range based on terrain. From the 3.5 DMG, visibility ranges in various terrains are something like:
Forest:
Dense forest: 2d6x10 ft
Medium forest: 2d8x10 ft
Sparse forest: 3d6x10 ft

Marshes:
Moor: 6d6x10 ft
Swamp: 2d8x10 ft

Hills:
Gentle hills: 2d10x10 ft
Rugged hills: 2d6x10 ft

Mountains:
Maximum is 4d10x10 from the peak

Desert:
6d6x 20 ft (limited by heat distortion)
Dunes reduce that to
6d6x10 ft

Plains:
6d6 x 40 ft

Skipping the Underwater table for obvious reasons.

A group of 1000 people is going to be large enough that when visibility is possible, at least one side spots the other.
I'll assume they're effectively around 300 ft radius- some people will travel together, others will travel more spread out, it's unlikely to be a perfect sphere but instead spread out in a line or oval, etc. 1000 people is a lot of people.

In contrast, the players have an effective radius of 0 ft (for simplicity).

Assuming that paths are independent (i.e. pretend they're moving perpendicular to each other across the area), the odds that people's paths intersect at a time where they're within visibility range is roughly
pi * (average visibility range + tribe radius)^2 / (total width of the square)^2
For Gentle Hills, the average visibility range is 110 ft, so the encounter odds for a single trip is about
pi * (410 ft)^2 / (50 miles)^2
=7.57721713 × 10-6
So about 1 in 100,000 per trip to randomly cross paths.

Arguably, we should use the largest possible visibility (200 ft), which would roughly increase this by about 50%.

This gets waaaay more common in the plains, with an average visibility range of 840 ft-
odds of 5.85803176 × 10-5 out of 1, or 0.000136471045 under the largest possible visibility (1440 ft).

Having 8 tribes brings this up by a factor of 8, but it's still improbable.

So it's unlikely to 'just run into' someone, even a group of people, if you're traveling at random.

However, people don't just travel at random, they travel along well-trodden paths, end their journeys at places with fresh water (rivers or lakes), and so on. The party might find traces of a tribes' past migrations, and decide to follow the easiest route.

If both parties are traveling along the same path at equal rates, there's a 50% chance they're going in opposite directions and are guaranteed to meet.
If they are traveling in the same direction, they probably have about a 1/4 - 1/3 chance to run into each other by chance- one stops for the night, and the other runs into them.
So the odds of running into each other if on the same route are something like 60% if both are taking the exact same route through the hex.

So, let's assume for a second that both parties are traveling in parallel lines (50% chance- maybe they zigzag, but 50% on average) and taking the 'best route' within about five miles (i.e. there are about 5-10 routes through the square, and each traveling group chooses one at random).

Then they have about a 50% chance to be traveling along a parallel route at the same time as you, 1/5-1/10 chance of taking the same route, and a 60% chance of spotting each other if on the same route- working out to about a 3-6% chance of running into each other on a trip, or about 0.75-1.5% per day.

If the tribes are immobile rather than migratory, the odds go up significantly further. There aren't rules on how much land it takes to support a single person, but it looks like in the range of 20 people per square mile (https://medium.com/migration-issues/notes-on-medieval-population-geography-fd062449364f) for small towns (which the 3.5 DMG classifies 1000 people as)- so each tribe's land is about 50 square miles, or about 7 miles square. That gives a fairly straightforwards estimate of a 14% chance for your travel route to go through a tribe's territory (i.e. a random route through the square hits the 7 mile span) - or about a 3.5% chance per day of travel.

Strictly speaking, I wouldn't expect them to all be in one group- there would probably be a couple smaller hamlets of 20-150 people and a larger 'central town' where people gather, but the total area covered might be about the same.

For multiple tribes, roll those odds separately for each trip or day.

[TheYell]

Thank you!