What a so-LOO-tion! Video reveals how to find the cleanest public toilet at a festival - using mathematical formulas
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If you've ever queued for a public toilet at a festival or event, you'll know how difficult it is to find the cleanest option, quickly.
With that in mind, a mathematician has devised a system of determining how many toilets must be rejected automatically to guarantee the best possible option.
When faced with a row of 100, her conclusion is that the first 37 should be avoided, and then each should be opened in order to make the correct choice.
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What's the best way to pick a public toilet at a festival or event? According to Dr Ria Symonds. for YouTube channel Numberphile. she claims that out of 100 toilets, people should discard the first 37 and then pick the next best toilet based on the 37 they've already observed, to get the cleanest experience
THE MATHS BEHIND THE PROBLEM
In the video, Dr Symonds draws three toilets on a piece of paper, and numbers them one to three - one being the most hygienic, and three being the dirtiest.
The order of these doors is then randomly mixed up.
There are six permutations of this problem, making it three factorial.
The chances of the first toilet being clean is slim, especially at large numbers so Dr Symnods suggests rejecting the first toilet in each permutation, and selecting the next best.
Using the first permutation, Dr Symonds can't pick the first, but the second isn't as clean as the first, so that leaves the third as her only option.
She then continues to go through the system in the other five permutations, and by rejecting the first each time, picks the cleanest toilet on three occasions. This results in a three out of six, or 50 per cent, chance, of picking the best toilet overall.
When this number raises to 100, she applies the same principle and discovers that the best way to get the cleanest possible toilet is to reject the first 37.
This particular mathematical problem is similar to the 'secretary problem'.
This famous conundrum asks how, out of 100 candidates, an interviewer would pick the best person for a job.
The same principle can also be used to select the best toilet at a festival, apparently.
'If you ever go to a music festival, especially a music festival in the UK, you might come across the problem that there's always lots of toilets for you to choose from,' said Dr Ria Symonds from the University of Nottingham in the video.
She explains that 'the problem with these toilets is they're not always that well looked after.'
So, in this particular problem, there can be considered three types of toilet: one that is deemed disgusting, one that is adequate, and one that it spotless.
'Ideally I'm looking for the best toilet, the most hygienic, the cleanest toilet, one with plenty of toilet paper in there and hand sanitiser and has been looked after,' she said.
When searching for the best toilet she imagines a row of many, in this case 100, and in the problem a festival-goer works through the toilets one-by-one.
But the second constraint is that you can't go back to a previous rejected toilet as 'there are people queieing behind you.'
After some painstaking maths, which you can watch in the videos below, Dr Symonds ultimately comes to her conclusion.
Glastonbury is famed for its rather unsanitary toilets. Here festival-goers are seen using the newly installed long drop toilets in the rain at the festival at Worthy Farm in Somerset on 26 June 2014. Perhaps with this new mathematical method, though, people won't need to seek these new toilets out
'If we were in Glastonbury and there were 100 toilets, what we should do is reject the first 37 toilets that we see,' explained Dr Symonds.
She clarified that you need to look in each of those toilets.
After that, you should then pick the next best toilet that you see as compared to the previous 37, which will get you the optimum standard of toilet.
This will prevent you from running out of toilets before finding one suitable, but it also makes sure you will find the best available toilet as quickly as possible.
And after the 37th toilet, you then also have a 37 per cent chance that a toilet you pick is actually the best category of toilet (lots of toilet paper and soap) in the remaining 67 per cent.
This method was also famously suggested to be used in the dating world to find the best partner - although perhaps lining up 100 men or women as prospective partners wouldn't go down so well.
THE MONTY HALL PROBLEM
If you're in the mood for more mathematical brain teasers you might also enjoy the famous Monty Hall problem, a paradox named after the host of the American TV game show Let's Make a Deal.
In the problem a contestant is presented with three doors: behind two is a goat and one has a new car - the latter of which they obviously want to win (contestants who would rather win one of the two goats are not considered here).
The contestant is then asked to pick one of the three doors but, before they are allowed to open it, the gameshow host (who knows the location of the car) opens one of the other doors, revealing a goat.
The contestant is then given the choice to switch. Should they switch, or should they stick with their door? Does it even matter?
It's intuitive to think that it doesn't matter as, with two doors, there is a 50:50 chance the contestant will pick the car.
But a bit of maths shows that switching is actually much more successful. In fact, the car will be chosen 2/3 times if the contestant switches, compared to just 1/3 if they don't.
The reason is explained in the image below. In each it is assumed the contestant chooses door 1 each time, although it doesn't matter which they actually pick - this is just an example.
As can be seen, when they make their initial selection they have just a 1/3 chance of being correct.
But after the host has revealed a goat, switching then gives them a 2/3 chance of getting the car.
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