2 Fundamentals of statistics and probability: Green and yellow gummy bears.

Slide 0

Welcome to the second episode of “Sparking conversations about math” series. Today we will talk about another experiment and in particular work through an important difference between dice and gummy bears. Now be honest: Would you have known what the two have in common? But let’s start from the beginning.

Slide 1

There are important problems and less important problems. Judge for yourself which category this “problem of the week” that I actually found on the Web falls into. Lucas wonders why there are so few green and red gummy bears in the bags. He likes them the best, but he more often finds yellow, orange, or – as he expresses himself – these stupid white gummy bears. Abena disagrees with him. She loves orange and yellow ones and yet finds them much less often than the red gummy bears.

Slide 2

Apparently we are dealing with a problem in which data is insufficient or perhaps even conflicting.

Does every bag of gummy bears contain a yellow gummy bear? Or even better, stated in general terms: Are there actually as many yellow gummy bears as green ones or gummy bears of any color in the bags?

Slide 3

Obviously, there’s no avoiding an experiment. And this is the basis: Gummy bears come in six colours, namely white, yellow, orange, light red, dark red, and green. 12 bags should be a reasonably good data base. We open the bags, resist the urge to pop one into our mouth and count.

Slide 4a

So, let’s open a bag. We determine how many gummy bears are even in a bag and of course count how many we find of each of the different colours. After the first bag not all colours are present. Let’s open the second bag. There are still a few red, also yellow, a few green, there is also orange and yes, there is a white gummy bear. Now all six colours are there. I have to arrange, that you can clearly see how many of which colour are there.

And we come to a conclusion: In fact, there are not many red gummy bears instead there are the most of the white ones. But let’s leave the intermediate results, let’s see what is emerged after twelve bags. So what is the final result of this data collection?

First of all, you can see that there were different numbers of gummy bears in the bags. The sum of all twelve bags is 102. It’s the bags g₁, g₂, g₃ etc. until g₁₂. You can write this formally. It’s the sum of all g_i equals 102 for i between 1 and 12.

$\sum _{i=1}^{\mathrm{12}}\mathrm{g_<em>i</em>\; =102}$

Looks really good mathematically, doesn’t it? If you divide 102 by 12 you get the mean m_i which equals 8.5. So there are on average 8.5 gummy bears in a bag.

On the left you see the state after 8 bags, in the middle the state after 10 bags and on the right the state after 12 bags. Obviously the dark red and green are not in the majority, but many white gummy bears were in the bags.

Slide 4b

In a table it looks like this. Looks very organized, doesn’t it? Although even the real gummy bears were divided into blogs of five which gave them a certain mathematically period.

Slide 5

How are the gummy bears distributed among the different colors?

You have seen that there are a total of 47 red, both light and dark red, and green gummy bears. By contrast, 33 are yellow and orange. However, if you add the 22 white ones to this group, then the count is 47 to 55. It actually makes a relatively fair impression.

But is that always the case? Does this distribution apply to any 12 bags? Have we “proven” the distribution with our experiment?

Slide 6

Not likely. What you see here are other distributions. However, they are based on somewhat smaller numbers. On the left the count is 9 to 16, so there is clearly a shortage of red and green gummy bears. On the right the count is 15 to 11, and red and green bears have the advantage. By the way, the distributions were used from real life and were therefore more or less easy to count.

Slide 7

Did you notice anything? Well, taking a gummy bear from a bag is almost a little like a rolling dice. In both cases, there are six possible results, namely either the numbers from 1 to 6 or the six different colors.

Is there also a difference between rolling a dice and taking a gummy bear from a bag?

Slide 8

Well, rolling a dice involves a theoretical assumption:

If you roll a non-manipulated dice n times, each of the events 1, 2, 3, 4, 5, 6 should occur approximately n/6 times when n is large.

Rolling a dice is a Laplace experiment. All results are equally probable.

There is no theoretical assumption for taking a gummy bear from a bag. There’s only one thing to do to determine the distribution: call the manufacturer, who should know.

Taking a gummy bear from a bag is NOT a Laplace experiment.

Slide 9

Numerous other examples of Laplace experiments exist. These include

• flipping a coin where heads or tails can occur;
• drawing numbers in a lottery where a ball is drawn from initially 49 balls;
• hitting a number between 0 and 36 in roulette, because here too the ball rolls without any sort of priorities;
• spinning an arrow and landing on one of the four colors of the wheel that you see at the right.

Other examples of random experiments that are not Laplace experiments are

• tossing a thumbtack, because although there are only two possibilities, they do not occur equally often;
• rolling a matchbox, where there are three basic possibilities with varying levels of difficulty to achieve.

I'm sure others come to mind. And you can very simply conduct many of them in practice.

Slide 10

That’s all for today. Thank you for listening and for your interest.