### Probability and Intuition

*[This is an automatic translation of the original post in Spanish and has not been edited yet.]*

Humans are not very good at calculating or making sense of the probability of uncertain events, an observation that has been demonstrated time and again in the context of understanding how rational we are as economic agents. Therefore, filling out the Panini Album of the Soccer World Cup is not only the perfect way to start to tune into the feeling of the tournament, but also an excellent exercise to show how fine-tuned my probabilistic intuition is.

Already in my previous post I had exposed my strategy to fill the album. In this post I share three questions that are easy to understand but that can be a bit more complex when solving them. In each one I made an estimate before doing the exact calculation to see how correct my intuition was, but in all cases I greatly underestimated the random dynamics that filling the album has.

1. How many envelopes will I open before the first repeat sheet comes out?

My intuition told me that it would be after opening about 10-15 envelopes. My memory was that one opened many envelopes with great emotion until after a while a repeated little monkey arrived that spoiled the good run. But this year I took care of the envelopes I bought, and it turned out that the first repeat came out in the sixth envelope!

Doing the exact mathematical calculation, it turns out that this observation is just normal: more or less one should expect that after opening 7 envelopes the first repeated sheet appears.

2. How many envelopes will I open before I get one whose five sheets I already have?

The feeling of frustration of opening an envelope whose five little monkeys come out repeated with the ones one already has is indescribable. It’s a mix of outrage and amazement, as if Panini has somehow managed to find a sophisticated way to rob us. My intuition was that a traumatic event like that should be strange, possibly something that happened after being well advanced in the collection, perhaps with more than 75% of the full album, that is after about 500 monkeys. This year when I did the real experiment, the fateful moment came to me long before, when I had only 355 monkeys.

The mathematical calculation confirmed that my intuition was confusing me: It can be predicted that on average after having opened some 74 envelopes, this is a total of 370 monkeys, one will come out in which the five sheets already have one.

3. If one does not exchange pictures, how many do you have to buy to complete the album?

Having intuition about this question is even more difficult because I think that neither me nor anyone else has ever thought of not exchanging monkeys when the album is filling up. Trying to answer it without doing any calculations, I estimated that maybe two or three times the total number of plates in the album were needed, this is between 1,200 and 2,000.

Now, although I am convinced of the virtues of doing real experimental tests, I am not willing to spend money on who knows how many envelopes of slides to prove this point. That’s why I preferred to do a simulation on the computer, where I can fill as many virtual albums as I want and I can better gauge my perception of the problem.

The figure below shows the result of simulating 1,000 albums, each with 680 plates. On the X axis is the number of envelopes that have been bought (each with five different monkeys between them) and on the Y axis is the number of sheets that are missing to complete the album. Each of the colored lines corresponds to the process of filling an album without making exchanges. Attention that the scale on the Y axis is logarithmic!

When no envelope has been opened, the 680 plates are missing, and that is why all the lines begin in the upper left corner, however, from that point on, each album has its own evolution with the envelopes that correspond to it. After opening about 140 envelopes (for a total of 700 little monkeys, a little more than the whole album has), there are still 250 sheets left to complete the album. And even reaching 400 open envelopes we see that all simulated albums are missing between 25 and 50 sheets. It is only after 565 envelopes that we see the first album with all the plates, but clearly that was an exceptional case of luck: on average the simulated albums took 972 envelopes (about 4,860 plates) to be completed. In the figure you can see a couple of unfortunates who needed more than 1,400 envelopes, the less fortunate requiring 8,915 monitas.

It is possible to do the exact calculation to answer this question without the need for a computer simulation. The math that is required is somewhat sophisticated if one wants to correctly model what Panini calls the “fifimatic” process, which he uses to randomly assign the five sheets that go to each envelope, guaranteeing that there are no repetitions between them. If one relaxes that assumption and assumes that there may be repeats in the same envelope, the mathematics are simplified and a closed formula can easily be found that predicts an average of 4,844 monkeys to fill the album if no exchanges are made.

But what happens in the slightly more realistic case in which exchanges with a group of friends are allowed? I will discuss that result on another occasion, as well as the details of the mathematics of this post.