Issue 027 - Box office - Attendance and capacity throughput
How much theater capacity did Spider-Man consume?
AP reported that Spider-Man: Brand New Day earned $360 million in U.S. and Canadian theaters during its opening weekend, beating the previous North American opening-weekend record held by Avengers: Endgame.
The problem
Estimate the total number of tickets sold for a $360 million North American opening weekend.
Then estimate the share of North American movie-theater capacity devoted to the film over the three-day weekend.
Use your estimate to determine total screenings required, combined audience-hours, and the approximate number of theater screens showing it at any given time.
Because Fermi problems target an order of magnitude, I normally use no more than two significant digits and write most calculations in scientific notation; the Fermi reference explains both conventions.
Before checking sources
Matt's first pass
I assumed the average ticket cost about $15, the average theater has a capacity of about 300 seats, and the average screening over the weekend was 90% filled, or about 270 people attending.
I also assumed that if a screen is dedicated to the film, it would show the film about 13 times over the weekend: 4 each day Friday, Saturday, and Sunday, plus one early Thursday midnight showing. I assumed the movie lasts about 2.5 hours.
tickets sold ~= $3.6 x 10^8 / $15
~= 2.4 x 10^7 tickets
~= 24 million ticketsThen:
screenings ~= 2.4 x 10^7 tickets / 270 people/screening
~= 9 x 10^4 screenings
screens ~= 9 x 10^4 screenings / 13 screenings/screen
~= 7 x 10^3 screensThat feels like a normal number of screens carrying a big blockbuster film in the U.S. and Canada, so maybe my ticket-price estimate is too low, meaning more tickets sold, or my estimate for people who can fit in a single screening is too low, meaning more people per screening.
If the film is 2.5 hours long and 24 million people saw it this weekend, then people spent:
audience-hours ~= 2.5 hours x 2.4 x 10^7 people
~= 6 x 10^7 person-hoursI converted that to about 70,000 years and about 870 human lifetimes, though that conversion needs checking.
Calibration Score
Matt's Calibration Score: 70 / 100
Higher is better: earn points for accurate pegs, sound models, correct math, and a result close to the sourced answer. The image shows percent full of it: 100 minus the Calibration Score.
Pegs: 10/30. Ticket, occupancy, and screen-schedule pegs were a bit high but usable.
Model: 30/30. Gross divided by ticket price, then screenings and screen capacity, was the right model.
Math: 0/10. The human-lifetime conversion had an order-of-magnitude mistake.
Result: 30/30. The core tickets, screenings, and screen-count estimates were very close.
Grounding facts
A 23-million-ticket weekend means something like one out of every fifteen people in the U.S. and Canada bought a ticket in just a few days, before accounting for repeat viewers and international audiences.
Theater count can hide the real constraint. A film playing in 4,487 theaters might still consume 10,000 or more screens if multiplexes dedicate several auditoriums to it all weekend.
After checking sources
Check and recalibrate
Matt's ticket estimate is strong. If average ticket price is $15 to $16, then:
tickets ~= $3.6 x 10^8 / ($15 to $16)
~= 2.3 x 10^7 to 2.4 x 10^7 ticketsSo a good answer is about 23 million to 24 million tickets.
The main adjustment is screen capacity. A 300-seat average auditorium with 90% occupancy is probably too high for an average across every showing, every market, late nights, early mornings, smaller auditoriums, and premium screens. A better Fermi range might be:
average seats per auditorium ~= 150 to 250
average occupancy ~= 60% to 80%
audience per screening ~= 100 to 200 peopleUsing 140 people per screening as a central estimate:
screenings ~= 2.3 x 10^7 tickets / 1.4 x 10^2 people/screening
~= 1.6 x 10^5 screeningsIf each dedicated screen can run about 12 to 13 showings across previews and the three-day weekend:
screen-equivalents ~= 1.6 x 10^5 screenings / 13
~= 1.2 x 10^4 screensA reasonable range is roughly 10,000 to 15,000 screen-equivalents. AP's 4,487-theater count makes that plausible: it averages to about 2 to 3 Spider-Man screens per theater, with big multiplexes showing it on many screens and smaller theaters showing it on fewer.
For audience time:
audience-hours ~= 2.3 x 10^7 people x 2.5 hours
~= 5.8 x 10^7 person-hoursConvert that to calendar years:
person-years ~= 5.8 x 10^7 hours / 8.8 x 10^3 hours/year
~= 6.6 x 10^3 years
80-year lifetimes ~= 6.6 x 10^3 / 80
~= 80 lifetimesMatt's audience-hour number was right, but the years/lifetimes conversion was about an order of magnitude high.
The record probably reflects a combination: very high ticket volume, premium ticket prices, strong occupancy, and a huge share of available screens. The $360 million gross was not just unusually full theaters or just expensive tickets. It required multiplexes across North America to devote a large chunk of their weekend screen-time to one movie.
Post-check reflection
Matt's reflection
I was a bit high on the estimated average theater occupancy, and probably also high on the percentage full, which lowered my screen count by a good amount.
I goofed on calculating the total number of years spent watching across all attendees, so I was off on human lifetimes by about an order of magnitude. Otherwise I was pretty much right on. Nothing major changes in my appreciation of this news item.
Recommended memory peg
Remember opening-weekend tickets ~= box-office revenue / average ticket price. For modern U.S./Canada blockbusters, $15 to $16 per ticket is a useful rough peg, and a dedicated screen can run about 12 to 13 showings across a long opening weekend for a 2.5-hour movie.
Reader results
Bars show how submitted estimates sort into the answer choices from the gut-check prompt.