How many passengers are in planes in the air at any given time in the U.S.A.?
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The key here is to think about what information you have (or can estimate) which can help you get to the answer. You need to be clear about your logic and assumptions, and be clear-minded enough to be able to sense-check the ultimate answer you arrive at.

First, I clarify the question – what is meant by “airplanes”? All planes, private jets, commercial flights only, military, fly-overs? And what is meant by “right now” – literally right this second? What is meant by “over the US”? Landmass only? Does this include ocean territory? What about Alaska and Hawaii?

I will assume that “airplanes” can be divided into four categories: commercial passenger, freight, small private and military. There will be other types (like charters and crop-dusters) but to simplify things I will leave them aside.

I assume that these flights are only those flying domestically or to an international destination (but still over the US). I will exclude planes flying over the US from one country to another.

I assume that “right this second” means 6.45pm.

I assume that “over the US” means over the landmass of the contiguous 48 states.

Then I look at each type of plane to look for drivers of the ultimate number.

For commercial passenger planes, I assume a good driver would be the number of airports in the USA. I assume there are:

10 giant airports
20 major airports
50 medium airports
100 small airports

Giant airports would have about 20 flights per hour taking off, 18 hours per day. So, about 360 flights per day. At any one time, perhaps 15 per cent of these flights would be in the air. I would assume that the average plane would contain 250 people. So, that’s 360 x 15% x 250 x 10 airports = 135 000 people.

Major airports would have about 10 flights per hour, 18 hours per day. Fifteen per cent would be in the air at any one time, with an average of 200 people on board. That’s 108 000 people.

Medium-sized airports would have about 10 flights per hour, 15 hours per day. Fifteen per cent would be in the air at any one time, with an average of 100 people on board. That’s 112 500 people.

Small airports would have about five flights per hour, 12 hours per day. Fifteen per cent would be in the air, with an average of 50 people on board. That’s 45 000 people.

Freight is trickier to estimate. I would estimate that there are an average of 100 cargo flights per major city in the US per day, and I’d estimate there are 200 major cities. Fifteen per cent of these flights would be in the air at any one time. They would have an average crew of five. That’s 15 000 people.

For small private planes, I’d say there is one private plane for every 500 head of population (300 million people), and that one per cent of these planes might be in the air at any one time, with an average of four people on board. That’s 600 000 x 1% x 4 = 24 000 people.

For military I am less sure. Let’s assume there are 10 000 military planes in the US (big and small). Let’s say five per cent are airborne at any one time with an average of 10 people aboard. That’s 5000 people.

Now I can add up all the people. The total equals 444 500 people in the air right now over the US.

To sense-check this, let me just divide by 200 (the average number on board a commercial airliner). That’s 2222 planes in the air, or 46 over each of the 48 states.

Hmm, this seems a bit on the high side, so I may have over-estimated somewhere. I can go back through my assumptions and calculations to see if I can trim the number a bit. However, I don’t think I’m far off.

by
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+1
Nimesh, very good answer indeed. I have a question for both you and Bijan, what is the basis of estimating the # of airports inside the US? It does not seem to correlate to any other facts.

Also how did you come up with the assumption that 15% of the planes are in the air at any time? Thanks a lot!

This will be variable based on the time of day, the day, and other factors. So let’s use some guesstimates.

A flight from a major hub leaves every 2 minutes. Lets say each flight is in the air for an average of 2 hours. There are 30 major hubs. 200 minor and medium hubs with flights leaving every 5 minutes.

Then do the math on how planes in the air at any given time…

Alternatively you could simplify and just assume a certain number of planes in the air but this shows less thought and you don’t call out where you are making assumptions or errors potentially:

500 large planes each holding 90% occupancy with an average of 200 passengers each.

1000 medium and small planes each holding 80% occupancy with an average of 100 passengers each.

Etc.

by
+1
I was thinking of taking a similar approach but my concern was there are two uneducated assumptions I have to make:
1- Number of airports in the US
2- Average time between each departure
3- Average time of flights
I’m also not considering the impact of foreign flights.
Is there a better way to make this estimate?

We are talking about the planes in the air. What different things do you see flying in the sky? ;)

You see

1. passenger planes
1. international flight with destination/origin in USA
2. national flights in USA
3.  flights crossing USA but not stopping there
2. military planes
3. private planes
4. spaceships
5. birds

Lets concentrate on passenger planes with a destination/origin in USA

Which cities can be an origin/destination? Those that have an airport. Which cities have an airport? Those that are big enough. Very few small cities have airports, except for very cheap airports which are in the middle of nowher, near a very small city.

So I will formulate the estimation for number (2) into this:

#flights with origin/dest in USA = #arrivals per airport * #airports + #departures per airport * #airport

I will break the problem into estimating for big cities and middle cities. Big cities have more than 3 million population. middle cities have an average of 400k population.

Big cities

Assumptions

• I will assume USA has 5 big cities with avg 2 airports.
• I assume each airport has 10 departure every 5 minutes, which makes 120 flights an hour. The same I assume for arrivals.
• I assume 60% of arrivals are from US airports, so they are already counted in departures. 40% are coming outside US. We assume that the distribution of departures/arrivals is the same, although possibly between midnight to 5 am there are less flights in certain cities, due to noise restrictions for example. Some airports do not have flights also 24hours.
• There are many flights with big planes, because there are more destinations, more passengers and the airport has the capacities and facilities for big planes. I assume 40% of flights are in big planes with 400 passengers, 20% middle planes with 200 passengers, 30% with 100 passengers, and 10% with 50 passengers.

#total flight in 1 airport in 1 big city = 120 departure + 120*40% arrival = 160 flights an hour

#passengers in one hour one airport= #passengers in big planes + #passengers in middle planes + #passengers in small planes +  #passengers in very small planes

= 160 ( 40% * 400 +20% * 200 + 30% * 100 + 10% * 100)=

= 160 X (160 + 40+ 30 + 10) ~160*600 = 40k passengers /hour

#passengers in 1 day in all big cities = #big cities * #airports * #passengers/day = 5 * 2 * 40k = 400k

----------------------------------

Middle cities

Assumptions

• I assume there are 50 cities with more than 400k people, and 1 airport each.
• I assume each airport has 1 departure every 5minute, which makes 12 flights an hour. The same I assume for arrivals. 40% of arrivals are from outside US, the rest is from USA, which is counted in departures. We assume that the distribution is the same, although possibly between midnight to 5 am there are less flights in certain cities, due to noise restrictions for example.
• There are NOT many flights with big planes, because there are less destinations, less passengers and the airport has less space for big planes. I assume 20% of flights are in big planes with 400 passengers, 30% middle planes with 200 passengers, and 20% with 100 passengers and 20% with 50 passengers.

#total flight in 1 airport in 1 big city = 12 departure + 12 * 40% arrival = 17 ~ 20 flights an hour

#passengers in one hour one airport= #passengers in big planes + #passengers in middle planes + #passengers in small planes

= 20 * 20% * 400 + 20 * 30% * 200 + 20 * 20% * 100 + 20 * 20% * 50 =

= 20 X (80 + 60 + 20 + 10) ~ 3.4k passengers /hour

#passengers in 1 hour in all middle cities = #middle cities * #airports * #passengers/hour = 50 * 1 * 3.4k =170k

====>

Total passengers in with origin/dest in USA in 1 hour = #passengers in 1 day in all middle cities  + #passengers in 1 hour in all big cities

= 400k + 170k = 570k

I assume one average flight is for 2 hours over the USA air space

International flights that land on very east or very west coast, are very briefly in the air of USA, e.g. 15 minutes. I assume 10% of flights can be ignored because of this. I assume 40% of flights are 1 hour, 30% of them are 2 hours, and 20% are than 2 hours.

However, I don't know how to adjust my calculations, so I skip this.

--------------------------------

Military planes

I assume there are 20 cities with military airports. There are 5 flights an hour. Each flight has 10 passengers.

#passengers on military planes in 1 hour = 5 * 20 * 10 = 1000 passengers

--------------------------------

private planes

I assume 1 out of 50k person has 1 private plane. USA has 300mio people.

#private planes = 300mio/50k = 6k

avg #capacity/private plane = 10

total capacity of private planes = 6kX10 = 60k

In every hour there are 5% private planes in the air.

===> total number of passengers in air in private plane = 60k * 5% = 3k

-------------------

total #passengers in the air = passenger in private plane + passengers in military planes + passengers in flights in USA

= 3k + 1k + 570k = 570k

Considering that some flights take more than one hour, I add 20% to this number.

===> total passengers = 570k*1.2 = 680 k passengers

According to Google, "At any given moment there are about 5,000 commercial airplanes in the sky over the United States"

Does 680k make sense?

If middle plane has 200 seats, that makes 680k/200 = 3400 planes in air

#planes over each state = 3400/50 = 70

which makes sense I think.

by (82 points)
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Hi Pegah,
I really like the structure of the answer and your ability to break it down to multiple segments based on some meaningful logic. Here are some feedback :
- Given there are different hours during the day and air traffic at 3am is very different than air traffic at 3pm, I suggest you pick a specific time of the day to do your calculations. The question does suggest "at any given time" but given the variety of time during the day, it's reasonable to make this adjustment. This will make it easier for you to determine which planes are in the air. For example, if you’re looking at all plans in the air at 3pm and average time of flights is 2 hours, you just need to look at # of planes that took off from 1pm to 3pm and multiple that number by # of passengers per plan (of course, adjust for the variation in size and international flights as well). You can apply the same model described above for other segments of the airline industry. If the interviewer insisted on "any given time", you can calculate the average number by using average number of planes that depart at any time of the day.
- I would count for empty seats by multiplying your number for passenger planes by something such as 80% (assume 20%of seats are empty
- For international flights, if average time in the US air for them is 15 minutes, then the number of international planes during a 2 hours divided by 8 gives you number of international planes in the air at a particular time

Assumption 1: Only passenger planes are in the air. (Its not a war time). And, ignoring personal, jet planes, helicopters.

Use cases: Domestic planes, International planes.

Case 1: Domestic planes: Assuming 50*50*2 terminals (50 states, 50 cities in each state, 2 domestic terminals on each airport).

Assumption 2: Assuming it takes 2 hours on an average to reach from 1 city to another in USA.

So, all the planes taking off from a terminal in last 2 hours will be in air.

Assumption 3: There are 2 runways on each terminal and it takes 10 mins to take off a plane.

So, on an average there will be 6*2*2 planes taking off in last 2 hours.

Assumption 4: Each plane carries 200 people.

So, the number of people in domestic planes in the last 2 hours: 200 (people)*24 (planes)*50*50*2(terminals).

Assumption 5: All the airports are operating at their full capacity. And, capacity of domestic and international airport is the same.

Hence, the number of people in the air is : p = 3*200 (people)*24 (planes)*50*50*2(terminals).

Here, 3 signifies for 1 domestic planes, 1 international planes flying inwards USA and 1 international planes flying outwards.

Here, if we assume that the airports are operating at x% capacity so the number will be p*x/100.

where, p is the total number of people calculated above.

by (44 points)