Matches

És casualitat o són matemàtiques?​

This 15-hole device allows you to check a surprising coincidence.
It is about randomly and independently putting the 8 balls in the holes. The best way to do itit is that 8 different people put each one ball without looking where the others have put it.
If we do not have about 8 people, we can use the bulwark or dice to have a random number.
Then the doors are raised and we will see how some balls have matched in the same hole.

  • Hotel Area: Sala Lluís Santaló
  • Minimum age: from 6 years old.
  • Required time: 5-10 minutes.
  • Number of participants: Ideally 8 people.
  • Keywords: birthdays, opposite event
  • Taxonomy: PROBABILITY

We often marvel that birthdays, deaths and other historical events coincide on the same date on the calendar. We also find it difficult to explain surprising coincidences of names, people, data or circumstances.
In this device, although there are envelope holes for all the balls, it is checked how chance leads to some balls matching in the same hole.

Explanation

In fact, it is quite possible that the 8 balls are all in different holes, what happens is that the probability of this happening is very low. Exactly 10%.

Per calcular aquesta probabilitat és convenient pensar en el succés contrari, és a dir, que cap bola coincideixi. Per calcular aquesta probabilitat, multipliquen la probabilitat de cada bola.  La primera bola té una probabilitat de 15 sobre 15 de no coincidir. La segona de 14 sobre 15, la tercera de 13 sobre 15, etc. El resultat de multiplicar les 8 fraccions és 0,101 o expressat en percentatge el 10%
Així si la probabilitat que totes les boles vagin a un forat diferent és del 10%, la probabilitat que això no passi i, per tant, alguna bola coincideixi és del 90%  

modul_coincidencies_formula

Aquest mòdul és una adaptació de la coneguda paradoxa dels aniversaris que consisteix en el fet sorprenent que en un grup de 50 o més persones la probabilitat que com a mínim dues persones coincideixin en el dia del seu aniversari és del 97%. 
En aquest cas ens hem d’imaginar que tenim 365 forats corresponents als dies de l’any i les boles són les persones situades en un d’aquests dies a l’atzar. 

The attached table shows the different values of this probability.  

Just as calculating the probability of a ball matching is easy, Calculating the odds that exactly two balls match, that exactly three balls match, etc.  is much more laborious.
You can download a spreadsheet with these counts.

Downloadable material