Puzzling Asked on October 4, 2021
Besides this problem I also use the following one to rise the interest of my students on the Probabilities Theory. It’s a funny puzzle were the use of simple arithmetic rules doesn’t work. We have to workaround a little bit to find the answer.
So here it goes:
The Random Family decided to go on a picnic next Sunday.
So they tried to get some information about the weather and predict if it will be a sunny Sunday.
They went online to weatheronsundays.org and happily saw that Sunday most likely be sunny (the algorithm used on this site forecasts correctly the weather about four in five times).
The youngest daughter also went to the CheckTheWeatherOnNextSundaySoYourFamilyPicnicWillBeASucess application on her smart phone and also saw that next Sunday will be sunny (this application forecasts correctly about three times out of four).Based on this two predictions, what is the probability of the next Sunday will be sunny?
(After rechecking my numbers which gave me a different answer before, it seems I've written a duplicate of hexomino's quicker answer. Since this one has 2 lines of maths instead of many, I think I'm going push post anyway.)
The key to figuring this out is that
This allows us to figure out the relative frequency of the two cases:
From this we get that whenever the two forecasts agree, it is
Correct answer by Bass on October 4, 2021
For these sorts of problems I like to use
In this scenario we can apply it as follows
Answered by hexomino on October 4, 2021
I've made this community wiki, so please edit away!
This is a very good question to put to students as long as one subsequently hammers home the point that it is ill posed and one works out the common fallacies that contribute to the expected answer.
The question is actually well suited for this didactic exercise because the tacit assumptions OP's preferred "simple" answer makes are quite implausible making it easier for students to appreciate that these assumptions are not a mere technicality but an actual mistake.
Assumption 1: p(sun) = p(no sun)
A good exercise for students to figure out where in the simple answer this assumption is hidden and how it impacts on the answer.
Also good to highlight the flaws of "no info: let's use a 'flat' ground truth". If we had asked in terms of "rain" vs "no rain" does this mean 50% rain probabiliy is a natural assumption? What if we had asked "sunny" "rainy" "neither" is it 33% each now?
Assumption 2: the two forecasts are independent
Again, good exercise to ask students where in the simple answer this assumption is used.
Unless we have a very low opinion of weather forecasting in general the assumption the forecasts are independent is not only unjustified, it is actually implausible as both forcasts are predictions presumably based on similar info and methodology.
Assumption 3: p(forecast correct|actual sun) = p(forecast correct|no sun) = p(forecast correct)
One last time challenge students to find where this assumption is used.
This is perhaps the most tricky thing to fully appreciate so be careful to well explain why there is a distinction between false positive and false negative rates and why they are often not the same.
Answered by Paul Panzer on October 4, 2021
We can just draw a cube like so:
The answer is...
Answered by Nautilus on October 4, 2021
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