With the World Cup just 3 months away, the best bit of the tournament build up is upon us – the Panini sticker album.

For those looking to invest in a completed album to pass onto grandchildren, just how much will you have to spend to complete it on your own? Assuming that each sticker has an equal chance of being found, this is a simple random number problem that we can recreate in Python.

This article will show you how to create a function that allows you to estimate how much you will need to spend, before you throw wads of cash at sticker boxes to end with a half-finished album. Load up pandas and numpy and let’s kick on.

In [1]:

import pandas as pd
import numpy as np

To solve this, we are going to recreate our sticker album. It will be an empty list that will take on the new stickers that we find in each pack.

We will also need a few variables to act as counters alongside this list:

Stickers needed
How many packets have we bought?
How many swaps do we have?

Let’s define these:

In [1]:

stickersNeeded = 682
packetsBought = 0
stickersGot = []
swapStickers = 0

Now, we need to run a simulation that will open packs, check each sticker and either add it to our album or to our swaps pile.

We will do this by running a while loop that completes once the album is full.

This loop will open a pack of 5 stickers and check whether or not it is featured in the album already. To simulate the sticker, we will simply assign it a random number within the album. If this number is already present, we add it to the swap pile. If it is a new sticker, we append it to our album list.

We will also need to update our counters for packets bought, stickers needed and swaps throughout.

Pretty simple process overall! Let’s take a look at how we implement this loop:

In [2]:

while stickersNeeded > 0:
    
        #Buy a new packet
        packetsBought += 1

        #For each sticker, do some things 
        for i in range(0,5):
            
            #Assign the sticker a random number
            stickerNumber = np.random.randint(0,681)
    
            #Check if we have the sticker
            if stickerNumber not in stickersGot:
                
                #Add it to the album, then reduce our stickers needed count
                stickersGot.append(stickerNumber)
                stickersNeeded -= 1

            #Throw it into the swaps pile
            else:
                swapStickers += 1

Each time you run that, you are simulating the entire album completion process! Let’s check out the results:

In [3]:

{"Packets":packetsBought,"Swaps":swapStickers}

Out[3]:

{'Packets': 939, 'Swaps': 4013}

939 packets?! 4013 swaps?! Surely these must be outliers… let’s add all of this into one function and run it loads of times over.

As the number of stickers in a pack and the sticker total may change, let’s define these as arguments that we can change with future uses of the function:

In [4]:

def calculateAlbum(stickersInPack = 5, costOfPackp = 80, stickerTotal=682):
    stickersNeeded = stickerTotal
    packetsBought = 0
    stickersGot = []
    swapStickers = 0


    while stickersNeeded > 0:
        packetsBought += 1

        for i in range(0,stickersInPack):
            stickerNumber = np.random.randint(0,stickerTotal)

            if stickerNumber not in stickersGot:
                stickersGot.append(stickerNumber)
                stickersNeeded -= 1

            else:
                swapStickers += 1

    return{"Packets":packetsBought,"Swaps":swapStickers,
           "Total Cost":(packetsBought*costOfPackp)/100}

In [5]:

calculateAlbum()

Out[5]:

{'Packets': 1017, 'Swaps': 4403, 'Total Cost': 813.6}

So our calculateAlbum function does exactly the same as our instructions before, we have just added a total cost.

Let’s run this 1000 times over and see what we can truly expect if we want to complete the album:

In [6]:

a=0
b=0
c=0

for i in range(0, 1000):
    a += calculateAlbum()["Packets"]
    b += calculateAlbum()["Swaps"]
    c += calculateAlbum()["Total Cost"]

{"Packets":a/1000,"Swaps":b/1000,"Total Cost":c/1000}

Out[6]:

{'Packets': 969.582, 'Swaps': 4197.515, 'Total Cost': 773.4824}

970 packets, over 4000 swaps and the best part of £800 on the album. I think we’re going to need some people to swap with!

Of course, as you run these arguments, you will have different answers throughout. Hopefully here, however, our numbers are quite close together.

Summary

In this article, we have seen a basic example of running simulations with random numbers to answer a question.

We followed the process of replicating the album experience and running it once, then 1000 times to get an average expectation. As with any process involving random numbers, you will get different answers each time, so through running it loads of times over, we get an average that should remove the effect of any outliers.

We also designed our simulations to take on different parameters such as number of stickers needed, stickers in a pack, etc. This allows us to use the same functions when World Cup 2022 has twice the number of stickers!

For more examples of random numbers and simulations, check out our expected goals tutorial.

Numpy Pandas Random Simulation