# How many policemen are in the US?

In a product management interview, you will have to estimate something arbitrary such as “How many balls will fit in an airplane?” or “How many policemen are in the US?”. We’ll work through this problem today to get an idea of how we will approach this problem.

First of all, realize that the interviewer is more interested in your approach than the final answer. You just need to be in the same order of magnitude to get it right.

Step 1) Ask clarifying questions. Is this full time policemen only? Or do you want everyone on the roster? Are we only talking about cops on patrol or desk workers too? What about University police? For now, let’s say its the regular force including patrolmen and desk workers.

Let’s write our equation:

#Of police officers = Total population of the US * Fraction of patrolmen per capita * number of total cops per patrolmen.

The total population of the US can be estimated to be 300 million

The fraction of patrolmen per capita is around 500. This is an estimate. It could be more or less but we’ll just say there is one patrol officer for every 500 people in an area.

The number of total cops per patrol men will be two. Why? This is also an estimate. I assume that for every policeman on the street, there is another that is working at their desk.

When you solve the equation, you get 300million/500*2 = 1.2 million police officers.

Does this answer make sense? According to a suvery in 2008, there are 1.1 million full-time officers. We hit the answer right on the money!

Follow the steps and be sure to write down your equations. The final answer does not matter as much as your logic of thinking and being able to clearly show your reasoning.

Looking forward to the future, the possibilities are endless.

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