Probabilistic Forecasts - Can you make more money using them?

A traditional weather forecast will suggest a certain course of action, but a probabilistic forecast will sometimes disagree. Here’s a mathematical example of how your pavement café could boost its profits by ignoring a threat of rain.

When the weather is fine, your café can sit extra people outside, as long as there is an extra waiter to serve them. It costs you £40 to employ the waiter for the afternoon, but if it is fine the outside tables will bring in an extra profit of £200. Even after the waiter’s wage, you still make an extra £160.

If you ask the waiter to work the shift but it rains, you make no extra profit, and you still have to pay the waiter £40. The probability of rain is P. Should you book the waiter to work?

Your choices can be summarised in this decision (or happiness) matrix:

Your expected extra profit without the waiter is £0.
Your expected extra profit with the waiter is (chance of no rain x profit) – (chance of rain x loss)
= (1-P) x £160 – P x £40
= £160 – (P x £200)

You should ask the waiter to work if that will make you more money, in other words, if

expected profit with waiter > expected profit without waiter
£160 – (P x £200) > £0

This happens if P < 0.80, or 80%

So you should book the waiter, even when the chance of rain is as high as 80%!

How will this affect your profits? A rival café has opened up next door, but they only book their waiter when the traditional forecast says it’ll be fine (and bear in mind this can still be wrong). The graph below uses actual weather data and forecasts (traditional and probabilistic) to show your extra profits (in red) versus your rival’s (in blue).

You stand to make about £5,000 more than your rival. Which is enough to put the froth on anyone’s cappuccino…

Related links
Probabilistic Forecasts Part 1
Met Office
ECMWF

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