The most at the feeder was forty and the flock was three hundred

# Self-authored for the EML case corpus (no external origin). A birdwatcher counts
# the finches at a garden feeder every hour for a week; the most ever present at
# once is 40, and the watcher concludes that the local flock is 40 birds. How
# many birds actually use the feeder is computed below.
#
# The reasoning is careful. Every count was made at the same time each hour; the
# highest count really was 40; a week of counts really is a good sample; and the
# intent is exactly 'how many birds are there'.
#
# A count at one moment sees only the birds present at that moment, and birds
# come and go. A ringing study measures the whole population instead: 60 birds
# are ringed in the first week, and of 50 caught in the second week 10 carry
# rings - so the ringed 60 are a fifth of the population, which is 300. The feeder
# never held more than 133 per mille of them at once.

40 => most_at_the_feeder_at_once
60 => ringed_in_the_first_week
50 => caught_in_the_second_week
10 => ringed_among_the_second_catch

int(ringed_among_the_second_catch * 1000 / caught_in_the_second_week) => ringed_share_of_the_second_catch_per_mille
int(ringed_in_the_first_week * caught_in_the_second_week / ringed_among_the_second_catch) => birds_in_the_population
birds_in_the_population - most_at_the_feeder_at_once => birds_the_busiest_moment_missed
int(most_at_the_feeder_at_once * 1000 / birds_in_the_population) => busiest_moment_share_of_the_population_per_mille
int(birds_in_the_population / most_at_the_feeder_at_once) => population_vs_busiest_moment_times

"most at the feeder at once      : " + str(most_at_the_feeder_at_once) + " birds" ^0
"ringed in the first week        : " + str(ringed_in_the_first_week) ^0
"caught in the second week       : " + str(caught_in_the_second_week) + ", of which " + str(ringed_among_the_second_catch) + " ringed" ^0
"ringed share of the second catch : " + str(ringed_share_of_the_second_catch_per_mille) + " per mille" ^0
"" ^0
"birds in the population         : " + str(birds_in_the_population) ^0
"busiest moment's share of them  : " + str(busiest_moment_share_of_the_population_per_mille) + " per mille" ^0
"birds the busiest moment missed : " + str(birds_the_busiest_moment_missed) ^0
"population vs busiest moment    : " + str(population_vs_busiest_moment_times) + " times, and a half" ^0
"" ^0

# ---- what the watcher verified ----

"the most-seen-at-once reasoning" ^0
"  counts : every hour for a week, at the same minute" ^0
"  highest count : " + str(most_at_the_feeder_at_once) + ", real" ^0
"  sample : a full week" ^0
"  intent : how many birds are there" ^0
"  facts wrong : 0" ^0
"  verdict : THE FLOCK IS " + str(most_at_the_feeder_at_once) + " BIRDS" ^0
"" ^0
"  counting carefully at fixed times is the part done right here," ^0
"  and it is why " + str(most_at_the_feeder_at_once) + " really is the most that were ever there at once" ^0
"" ^0

# ---- how many birds use the feeder ----

"a snapshot and a census" ^0
"  what a count sees : the birds present at that moment" ^0
"  what the birds do : come and go, each spending a small part of" ^0
"    the day at the feeder" ^0
"  what ringing measures : the population as a whole - the ringed" ^0
"    birds mix back in, and the share of rings in a later catch is" ^0
"    their share of everyone" ^0
"  the share : " + str(ringed_among_the_second_catch) + " of " + str(caught_in_the_second_week) + ", " + str(ringed_share_of_the_second_catch_per_mille) + " per mille, so the " + str(ringed_in_the_first_week) + " ringed are a fifth" ^0
"  the population : " + str(birds_in_the_population) + " birds" ^0
"" ^0

# ---- what the watcher got ----

"the estimate" ^0
"  believed : " + str(most_at_the_feeder_at_once) + " birds" ^0
"  actual : " + str(birds_in_the_population) + ", of which the busiest moment saw " + str(busiest_moment_share_of_the_population_per_mille) + " per mille" ^0
"  is the count wrong : no; " + str(most_at_the_feeder_at_once) + " were there at once" ^0
"  is the most at once the number there are : no; it is the" ^0
"    number that happened to overlap" ^0
"" ^0

# ---- null control ----

# The same flock estimated from the share of ringed birds instead of from the
# busiest moment at the feeder.
40 => nc_flock_read_from_the_busiest_moment
300 => nc_flock_read_from_the_ringed_share
260 => nc_birds_the_ringing_reveals

"null control - estimate from the ringed share" ^0
"  flock, read from the busiest moment : " + str(nc_flock_read_from_the_busiest_moment) ^0
"  flock, read from the ringed share : " + str(nc_flock_read_from_the_ringed_share) ^0
"  birds the ringing reveals : " + str(nc_birds_the_ringing_reveals) ^0
"  no bird and no feeder changed; the population was measured by" ^0
"  a method that sees the ones not present at the moment" ^0
"" ^0

# ---- the rule ----

"what a week of careful hourly counts guarantees" ^0
"  the most ever present at once was " + str(most_at_the_feeder_at_once) + " : exactly" ^0
"  the flock is " + str(most_at_the_feeder_at_once) + " birds : not addressed; birds come and go, and" ^0
"    the ringed " + str(ringed_in_the_first_week) + " making up " + str(ringed_share_of_the_second_catch_per_mille) + " per mille of a later catch puts the" ^0
"    population at " + str(birds_in_the_population) ^0
"" ^0

"a crowd at one moment is a sample of the crowd over a day; the busiest" ^0
"minute tells you how many overlap, and the rings tell you how many there are" ^0
"" ^0

"The most at the feeder at once was " + str(most_at_the_feeder_at_once) + " - the counts are right. But birds" ^0
"come and go: " + str(ringed_among_the_second_catch) + " of " + str(caught_in_the_second_week) + " birds caught in the second week carried the first" ^0
"week's rings, so the " + str(ringed_in_the_first_week) + " ringed are a fifth of a population of " + str(birds_in_the_population) + ", and the" ^0
"busiest moment saw " + str(busiest_moment_share_of_the_population_per_mille) + " per mille of it, until the flock is counted by its rings." ^0

most_at_the_feeder_at_once = 40
ringed_in_the_first_week = 60
caught_in_the_second_week = 50
ringed_among_the_second_catch = 10
ringed_share_of_the_second_catch_per_mille = int(ringed_among_the_second_catch * 1000 / caught_in_the_second_week)
birds_in_the_population = int(ringed_in_the_first_week * caught_in_the_second_week / ringed_among_the_second_catch)
birds_the_busiest_moment_missed = birds_in_the_population - most_at_the_feeder_at_once
busiest_moment_share_of_the_population_per_mille = int(most_at_the_feeder_at_once * 1000 / birds_in_the_population)
population_vs_busiest_moment_times = int(birds_in_the_population / most_at_the_feeder_at_once)
print("most at the feeder at once      : " + str(most_at_the_feeder_at_once) + " birds")
print("ringed in the first week        : " + str(ringed_in_the_first_week))
print("caught in the second week       : " + str(caught_in_the_second_week) + ", of which " + str(ringed_among_the_second_catch) + " ringed")
print("ringed share of the second catch : " + str(ringed_share_of_the_second_catch_per_mille) + " per mille")
print("")
print("birds in the population         : " + str(birds_in_the_population))
print("busiest moment's share of them  : " + str(busiest_moment_share_of_the_population_per_mille) + " per mille")
print("birds the busiest moment missed : " + str(birds_the_busiest_moment_missed))
print("population vs busiest moment    : " + str(population_vs_busiest_moment_times) + " times, and a half")
print("")
print("the most-seen-at-once reasoning")
print("  counts : every hour for a week, at the same minute")
print("  highest count : " + str(most_at_the_feeder_at_once) + ", real")
print("  sample : a full week")
print("  intent : how many birds are there")
print("  facts wrong : 0")
print("  verdict : THE FLOCK IS " + str(most_at_the_feeder_at_once) + " BIRDS")
print("")
print("  counting carefully at fixed times is the part done right here,")
print("  and it is why " + str(most_at_the_feeder_at_once) + " really is the most that were ever there at once")
print("")
print("a snapshot and a census")
print("  what a count sees : the birds present at that moment")
print("  what the birds do : come and go, each spending a small part of")
print("    the day at the feeder")
print("  what ringing measures : the population as a whole - the ringed")
print("    birds mix back in, and the share of rings in a later catch is")
print("    their share of everyone")
print("  the share : " + str(ringed_among_the_second_catch) + " of " + str(caught_in_the_second_week) + ", " + str(ringed_share_of_the_second_catch_per_mille) + " per mille, so the " + str(ringed_in_the_first_week) + " ringed are a fifth")
print("  the population : " + str(birds_in_the_population) + " birds")
print("")
print("the estimate")
print("  believed : " + str(most_at_the_feeder_at_once) + " birds")
print("  actual : " + str(birds_in_the_population) + ", of which the busiest moment saw " + str(busiest_moment_share_of_the_population_per_mille) + " per mille")
print("  is the count wrong : no; " + str(most_at_the_feeder_at_once) + " were there at once")
print("  is the most at once the number there are : no; it is the")
print("    number that happened to overlap")
print("")
nc_flock_read_from_the_busiest_moment = 40
nc_flock_read_from_the_ringed_share = 300
nc_birds_the_ringing_reveals = 260
print("null control - estimate from the ringed share")
print("  flock, read from the busiest moment : " + str(nc_flock_read_from_the_busiest_moment))
print("  flock, read from the ringed share : " + str(nc_flock_read_from_the_ringed_share))
print("  birds the ringing reveals : " + str(nc_birds_the_ringing_reveals))
print("  no bird and no feeder changed; the population was measured by")
print("  a method that sees the ones not present at the moment")
print("")
print("what a week of careful hourly counts guarantees")
print("  the most ever present at once was " + str(most_at_the_feeder_at_once) + " : exactly")
print("  the flock is " + str(most_at_the_feeder_at_once) + " birds : not addressed; birds come and go, and")
print("    the ringed " + str(ringed_in_the_first_week) + " making up " + str(ringed_share_of_the_second_catch_per_mille) + " per mille of a later catch puts the")
print("    population at " + str(birds_in_the_population))
print("")
print("a crowd at one moment is a sample of the crowd over a day; the busiest")
print("minute tells you how many overlap, and the rings tell you how many there are")
print("")
print("The most at the feeder at once was " + str(most_at_the_feeder_at_once) + " - the counts are right. But birds")
print("come and go: " + str(ringed_among_the_second_catch) + " of " + str(caught_in_the_second_week) + " birds caught in the second week carried the first")
print("week's rings, so the " + str(ringed_in_the_first_week) + " ringed are a fifth of a population of " + str(birds_in_the_population) + ", and the")
print("busiest moment saw " + str(busiest_moment_share_of_the_population_per_mille) + " per mille of it, until the flock is counted by its rings.")

most at the feeder at once      : 40 birds
ringed in the first week        : 60
caught in the second week       : 50, of which 10 ringed
ringed share of the second catch : 200 per mille

birds in the population         : 300
busiest moment's share of them  : 133 per mille
birds the busiest moment missed : 260
population vs busiest moment    : 7 times, and a half

the most-seen-at-once reasoning
  counts : every hour for a week, at the same minute
  highest count : 40, real
  sample : a full week
  intent : how many birds are there
  facts wrong : 0
  verdict : THE FLOCK IS 40 BIRDS

  counting carefully at fixed times is the part done right here,
  and it is why 40 really is the most that were ever there at once

a snapshot and a census
  what a count sees : the birds present at that moment
  what the birds do : come and go, each spending a small part of
    the day at the feeder
  what ringing measures : the population as a whole - the ringed
    birds mix back in, and the share of rings in a later catch is
    their share of everyone
  the share : 10 of 50, 200 per mille, so the 60 ringed are a fifth
  the population : 300 birds

the estimate
  believed : 40 birds
  actual : 300, of which the busiest moment saw 133 per mille
  is the count wrong : no; 40 were there at once
  is the most at once the number there are : no; it is the
    number that happened to overlap

null control - estimate from the ringed share
  flock, read from the busiest moment : 40
  flock, read from the ringed share : 300
  birds the ringing reveals : 260
  no bird and no feeder changed; the population was measured by
  a method that sees the ones not present at the moment

what a week of careful hourly counts guarantees
  the most ever present at once was 40 : exactly
  the flock is 40 birds : not addressed; birds come and go, and
    the ringed 60 making up 200 per mille of a later catch puts the
    population at 300

a crowd at one moment is a sample of the crowd over a day; the busiest
minute tells you how many overlap, and the rings tell you how many there are

The most at the feeder at once was 40 - the counts are right. But birds
come and go: 10 of 50 birds caught in the second week carried the first
week's rings, so the 60 ringed are a fifth of a population of 300, and the
busiest moment saw 133 per mille of it, until the flock is counted by its rings.

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