2021-04-23 53 9

Location

on a sports ground in Rellingen, Schlewig-Holstein

Participants

π π π (talk)

Expedition

Incredible – a hash only 384 m from 2021-04-10 53 9, 13 days previously! How unlikely is that‽ Well, according to Kripakko's calculations (see below and on the talk page), this happens on average once every 3.33 years. However, I don't visit every hash, so the probability of it happening for one of the geohashes I've been to this far (and could visit again) is only 6.1%. That's actually more than I expected! For equal odds of it happening I'd have to do 840 hashes, which at my current pace would take me about 23.5 years total!

Anyway, I took the train to Schnelsen in the afternoon and cycled almost the exact same route to the hash as 13 days before. I arrived at the sports grounds after 2.8 km, and discovered to my relief that they were public and not fenced off. However, I didn't expect there to be actual people training on them! I guess Covid restrictions must have been partially lifted recently. The hash lay on a grass football field right behind a new artificial turf field, and there was a football team warming up by running. I cycled to the edge of the field and pondered what to do, then decided to just walk to the hash and try to explain if somebody approached me. So I walked 55 m, found the spot, took pictures and awkwardly stood there while the players jogged right past me, less than 10 m away, loudly wondering about but not confronting me. I saw the coach pace toward me and got ready to explain myself, but he too strolled right past me to the edge of the field and then back (perhaps he'd changed his mind while approaching me). I quickly walked back to my bike, musing about the hilarious situation I'd just found myself in, and took off again.

I only had a few minutes tine, but I wanted to try to get as close to the hash from two weeks ago as possible. So I tried to get to the street in the west along which the tree nursery had been, but I couldn't find a path, so I just cycled to the edge of another sports ground that was adjacent to the forest through which I'd accessed the hash, 140 m from the coordinates. I noticed I only had 10 minutes left for the 10-minute ride, so I set off and exited the sports fields at the street where I'd met Fippe 13 days before, less than 100 m from that location. I didn't particularly race back, but still managed to be at the station four minutes early, giving me time to eat some cookies before catching the train home.

Kripakko's calculations

taken from the talk page – see there for more discussions about the calculations and methods of computing areas on a spherical or spheroidal Earth

Consider a circle of radius r at latitude θ and longitude ϕ. In the latitude/longitude coordinate space, if the circle is small (r ≪ graticule dimensions), it will look like an ellipse with semi-major axis ϕ_r and semi-minor axis θ_r. The semi-major axis will always be longitudinal due to the non-square shapes of the graticules. The lengths of the axes are:

ϕ_r = 1° × r / w_grat,

θ_r = 1° × r / h_grat,

where w_grat and h_grat are the width and the height of the graticule respectively at the centre point of the ellipse. Assuming that the Earth is a sphere with radius R = 6371000 m and taking θ = 53.6401767° from 2021-04-10_53_9, the dimensions of the graticule are approximately

h_grat = π R / 180 = 111195 m,

w_grat = h_grat cos θ = 65922 m.

Now, if r = 384 m, we get the axes of the ellipse:

ϕ_r = 0.00583°,

θ_r = 0.00345°.

The probability P₁ that a random hashpoint hits this ellipse is simply the area of the ellipse divided by the area of the graticule:

P₁ = (π ϕ_r θ_r) / (1° × 1°) = 6.32 × 10⁻⁵ = 1 / 15800.

Finally, the probability P₂ that a hashpoint hits the ellipse at least once in 13 days is

P₂ = 1 − (1 − P₁)¹³ = 8.21 × 10⁻⁴ = 1 / 1220.

An event of this probability will happen on average about once in 3.33 years or three years and four months.