# 3.1 Using the Fermi method

Here is an example of using Fermi’s approach to estimate the number of doctors in the UK who are general practitioners (GPs).

Assume that:

- Each GP spends 20 hours a week in individual appointments with patients. This seems a reasonable estimate because GPs have many other duties.
- Each GP works 44 weeks a year, allowing for holidays.
- Each individual appointment takes 10 minutes.
- The average person sees their doctor three times a year.
- There are 65 million people in the UK.

The 65 million people will between them need 65 million times 3 appointments, giving 195 million appointments.

Each appointment lasts 10 minutes, so the total time is 195 million times 10 minutes, or 1950 million minutes.

Each GP spends 20 hours a week on individual appointments, for 44 weeks. This gives a total of 44 times 20 hours, or 880 hours. Multiplying by 60 gives 5280 minutes.

Now, dividing the number of patient minutes per year by the number of GP minutes per year gives an estimate of the number of GPs needed.

Here is the complete calculation in Python.

>>> 65e6*3

195000000.0

>>> _*10

1950000000.0

>>> patient_minutes = _

>>> 44*20

880

>>> _*60

52800

>>> gp_minutes = _

>>> patient_minutes/gp_minutes

36931.818181818184

>>> round(_,-3)

37000.0

>>>

When the total number of patient minutes has been found, it needs to be stored while the number of GP minutes is calculated.

In Python, you can do this by attaching a label to a result. The label can be anything but using patient_minutes is a good choice because it makes it obvious what it represents. Similarly, gp_minutes is a good choice for the GP minutes.

Finally, the result is rounded to the nearest thousand GPs, giving an estimate of 37,000.

For comparison, at 31 March 2016, the GP headcount for England alone was 41,877. But many doctors were part-time, and the full-time equivalent was 34,914. This would scale up to about 43,000 for the whole UK.

So, the estimate of 37,000 is slightly low, but remarkably close, considering how many of the assumptions were no more than educated guesses.

Now try to work out a problem using the Fermi approach.

## Activity 6 Solving a Fermi problem

Estimate how many cats there are in the UK. (Remember: this is a rough estimate. You are not trying to get an exact count!)

You will need to consider:

- How many people live in the UK.
- Roughly how many people live in an average household. This will need to be a guess.

From these figures, you can estimate how many households there are in the UK.

Now:

- Guess what fraction of households have one or more cats.
- Households with cats often own more than one, so guess an average figure for the number of cats in a cat-owning household.

By combining these estimates

- how many households
- what fraction own one or more cats
- the average number of cats per cat-owning household

you should be able to arrive at a ballpark estimate of the UK cat population.

### Discussion

We assumed:

- UK population 65 million
- 3 people per household
- 1 household in 8 owns cats
- the average number of cats in a cat-owning household is 1.5.

Here is our calculation, rounding to the nearest 100,000 cats!

>>> 65e6

65000000.0

>>> _/3

21666666.666666668

>>> _/8

2708333.3333333335

>>> _*1.5

4062500.0

>>> round(_,-5)

4100000.0

>>>

So, there are just over 4 million cats.

Of course, your estimates are probably slightly different, but your result for Activity 6 was probably not that different from ours. Your guesses might have been better than ours because, according to Statista (2018), the number of cats was 8 million.

The next section looks at a different method of estimation, in which there are a large number of people to estimate a quantity and then take the average of their individual estimates.