R15 Probabilities

[Oggm]

Worked out a few probabilities this morning, they assume that only 'normal' R15s exist, with 6 rounds in a season, there are 134 possible stations to be picked, and that 20 of those lie south of the Thames.

Listed in increacing order of rarity:

All stations on the list are north of the Thames: 1 in 13.1 (correct to 1 d.p.)
Same starting station is picked twice in a season (six events): 1 in 13.4
Any two selected starting stations are the same: 1 in 134
All stations on list are in Zone 1: 1 in 79064.5 (correct to 1 d.p.)
All stations on the list are on the same line: 1 in 1,461,095,149 (courtesy of RJSRdg)
All the starting stations in a season are the same: 1 in 322417936
All the starting stations in one season are the same starting stations in the next season (order does matter): 1 in 43204003424
The stations in the list are in alphabetical order: 1 in 1307674400000 (correct to 8 s.f.) <- Came up in discussion last R15
All stations on the list are south of the Thames: 1 in 2376477700000000 (correct to 8 s.f.)
A season contains two rounds with the same list (order does not matter): 1 in 2733899905907677760
Any two selected rounds contain the same list (order does not matter): 1 in 27338999059076777600
A season contains two rounds with the same list (order does matter): 1 in 3575050831633081981155655680000
Any two selected rounds contain the same list (order does matter): 1 in 35750508316330819811556556800000

I will be happy to add any other probablilties anyone might wish for, workings out are avaliable on request, please notify me of mistakes when/if you find them.

[greatkingrat]

How about the probability all stations are in zone 1?

[Oggm]

How about the probability all stations are in zone 1?

Done, thank you, I have a feeling that there are infact 135 stations in zones 1 and 2, not 134 - embarassing mistake, as a result, everything is wrong, eeeek

[RJSRdg]

Probability that all 15 stations lie on the same line?

[RJSRdg]

Worked it out myself:

I in 1,461,095,149!

(The probability of all stations landing on any one of the Metropolitan, Victoria and of course Waterloo & City is zero).

District and Northen cheat slightly of course owing to their different branches, so whilst all stations may lie on the District or Northern Lines it doesn't necessarily mean you can visit them all simply by staying on the same train.
And in some cases, particularly on the Circle Line, staying on the same train may not be the quickest way of completing it.

[Oggm]

For this post:
Dis. = Number of District line stations in zones 1 and 2
Cir. = Number of Circle line stations in zones 1 and 2
And so on

Worked it out myself:

I in 1,461,095,149!

(The probability of all stations landing on any one of the Metropolitan, Victoria and of course Waterloo & City is zero).

District and Northen cheat slightly of course owing to their different branches, so whilst all stations may lie on the District or Northern Lines it doesn't necessarily mean you can visit them all simply by staying on the same train.
And in some cases, particularly on the Circle Line, staying on the same train may not be the quickest way of completing it.

Thank you, what was your method for working this out? I would have done 1/( (Dis.C15 + Cir.C15 + Cen.C15 + Bak.C15 + Ham.C15 + Nor.C15 + Pic.C15 + Jub.C15)/135C15 ), but It gave me a different answer - probably a mistake in counting stations

[RJSRdg]

Taking the Bakerloo Line as an example:
18 of the 135 stations in Zone 1 or 2 are on the Bakerloo Line.
So the probability of the first randomly selected station being on the Bakerloo Line is 18/135.
If the first station is on the Bakerloo Line, the probability of the second station also being on the Bakerloo Line is 17/134 as there are now 134 stations in Zones 1&2 to choose from and 17 remaining stations on the line.
So on right down to the 15th station - the probability of this being on the Bakerloo Line is (18-14)/(135-14)= 4/121.

Then to get the chances of all 15 stations being on the Bakerloo Line, multiply together

(18/135)x(17/134)x(16/133)x(15/132)x(14/131)x(13/130)x(12/129)x(11/128)x(10/127)x(9/126)x(8/125)x(7/124)x(6/123)x(5/122)x(4/121).

Then repeat the process for the other lines (except for the Metropolitan, Victoria and Waterloo & City, which don't have enough stations in Zones 1 & 2), and add together.

Full working at https://docs.google.com/spreadsheets/d/ ... ingle=true


An interesting thought I'm now having regarding the Circle Line - if Hammersmith is the starting station, and there are stations up the branch as far as Edgware Road, with none along the top of the Circle Line beyond that, how many and how far would there need to be on the western and southern edges of the Bottle for it to be worth staying on the same train and completing a full circuit rather than changing and reversing at Edgware Road?

[Lunaticonthegrass]

I have a feeling that there are infact 135 stations in zones 1 and 2, not 134 - embarassing mistake, as a result, everything is wrong, eeeek

Er, I thought there were 137 stations in Zones 1 and 2. (135 was pre-2021.)

[Oggm]

I have a feeling that there are infact 135 stations in zones 1 and 2, not 134 - embarassing mistake, as a result, everything is wrong, eeeek

Er, I thought there were 137 stations in Zones 1 and 2. (135 was pre-2021.)

All I now know after meticulously counting every station, is that there are 136, you might be right, I hope you are right, but so far I honestly have no idea. Please someone come rescue me!

[RJSRdg]

Those two extra stations take the chances of 15 stations on the same line to 1 in 1,844,135,267!

[Lunaticonthegrass]

I have a feeling that there are infact 135 stations in zones 1 and 2, not 134 - embarassing mistake, as a result, everything is wrong, eeeek

Er, I thought there were 137 stations in Zones 1 and 2. (135 was pre-2021.)

All I now know after meticulously counting every station, is that there are 136, you might be right, I hope you are right, but so far I honestly have no idea. Please someone come rescue me!

OK, after counting twice myself, I also believe that there are 136 stations in Zones 1 and 2. (I count 69 stations in Zone 2 not including those on the Zone 1/2 boundary, and I know that there are 67 stations in Zone 1.) Any advances on this?

[greatkingrat]

There are 70 stations in zone 2 (not including the zone 1+2 stations). Did you forget to count Hammersmith twice? Or maybe you missed Kentish Town which is currently closed?

Arsenal
Barons Court
Belsize Park
Bermondsey
Bethnal Green
Bow Road
Brixton
Caledonian Road
Camden Town
Canada Water
Canary Wharf
Chalk Farm
Clapham Common
Clapham North
East Acton
Finchley Road
Finsbury Park
Fulham Broadway
Goldhawk Road
Hammersmith
Hammersmith
Highbury & Islington
Holland Park
Holloway Road
Kensal Green
Kensington (Olympia)
Kentish Town
Kilburn
Kilburn Park
Ladbroke Grove
Latimer Road
Maida Vale
Mile End
Mornington Crescent
Oval
Parsons Green
Putney Bridge
Queen's Park
Ravenscourt Park
Royal Oak
St John's Wood
Shepherd's Bush
Shepherd's Bush Market
Stamford Brook
Stepney Green
Stockwell
Swiss Cottage
Tufnell Park
Warwick Avenue
West Brompton
West Hampstead
West Kensington
Westbourne Park
White City
Whitechapel
Wood Lane
Archway
Bromley-by-Bow
Canning Town
Clapham South
East Putney
Hampstead
Manor House
North Acton
North Greenwich
Stratford
Turnham Green
West Ham
Willesden Green
Willesden Junction

[miklcct]

My data count is 137, with 67 in zone 1 and 75 in zone 2. 117 are on the north of the river and 20 are on the south of the river.

Kentish Town is in my data set.

The source data can be downloaded here.

[Lunaticonthegrass]

There are 70 stations in zone 2 (not including the zone 1+2 stations). Did you forget to count Hammersmith twice? Or maybe you missed Kentish Town which is currently closed?

I just counted twice again and both times got 69 stations, definitely counting both Hammersmiths and not missing out Kentish Town. I then realised that I had missed Olympia on every count!

So it appears that 137 is the correct number.

[Oggm]

Thank you, 137 it is, I will update the corrected numbers when I find the time, sorry for the kerfuffle.