3 What exactly is stochastics? The somewhat different mathematics.

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What exactly is stochastics? In case you have always wanted to know this, you’re in the right place.

Slide 1

The term stochastics is derived from the Greek word “stochasmos”, which means “guess”. Stochastics comprises two subtopics and an important pillar.

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And – contrary to many a guess – neither the subtopics nor the pillar is mystical. See for yourself.

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Subtopic no. 1 is probability theory.

Probability theory involves generating and using suitable models for random events. Modeling is one core of mathematics … thus it is a rock-solid component of stochastics.

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Subtopic no. 2 is statistics. Here the focus is on measurement, that is, data collection, and the suitable representation of data. If you do this honestly, then it is certainly a rock-solid component of stochastics.

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Finally, combinatorics is a pillar of stochastics. Combinatorics involves counting; nothing is guessed. It is an important and rock-solid pillar of stochastics. By the way, combinatorics belongs to the main idea of “number” in educational standards and is therefore really a pillar and not a subtopic.

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Let’s talk briefly about the development of stochastics. No, this isn’t going to be a historical account at high speed. But it is indeed highly interesting that the origins of dealing with the topic go all the way back to ancient times, and these origins can essentially be traced back to games of chance, which were already popular then.

However, it was difficult that probability without experiments is inconceivable. In contrast, science in ancient times was based completely on logical argumentation. Mathematics in particular was characterized by this approach, which excluded experiments.

Do you remember that the relationship of the volumes of a cylinder with a radius and height of r, of a hemisphere with a radius of r and of a circular cone with a radius and height of r is of 3:2:1? The volume of the cylinder is the area of the base x height, thus πr²•r = πr³. The volume of the hemisphere is 2/3•πr³. The volume of the cone is 1/3•πr³. In schoolbooks you can also find a scale that is used to empirically verify the relationship. Completely unthinkable in antiquity.

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Certainly it is just as interesting that in the 16th century, a systematic treatment of probability and combinatorics was documented by Gerolamo Cardano and his book Liber de Ludo Aleae, Book on Games of Chance.

By the way, this is precisely the scientist who was the first to make use of complex numbers in solving cubic equations.

Otherwise, correspondence between Blaise Pascal and Pierre de Fermat in the 17th century is considered the birth of modern stochastics.

Ultimately, it was the Russian Andrey Nikolaevich Kolmogorov who initiated the “mathematical breakthrough” with his axiomatic system of probability, albeit not until 1933. 

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Naturally there are also problems with stochastics, precisely the question of when one should learn it. We want to consider three aspects below.

First, we will talk about the necessity of experiments, without which stochastics cannot work.

Then we will address intuition, which not infrequently contradicts mathematical reality, even among adults.

Finally, we will take a quick look at misconceptions. Especially among children, there are very specific ideas about stochastics that are unfortunately not always correct. Didactic research in mathematics deals with this quite successfully.

Slide 9

Let’s look at an example.

In the lottery, each time a bonus number between 0 and 9 is drawn. Now, this number was a 4 for three draws in a row. What is the probability that the number 4 will again be the bonus number for the next draw? 

What do you think? Mull that over for a moment.

Well, random events do not have memory. The probability is and remains 1/10 for each number and each draw.

However, faulty conceptions often exist. And it is exciting that they can be present in both directions. Most people polled estimate the probability of a 4 at the next draw lower compared to another number (“since a 4 occurred for the last three draws”).

However, some people argue that the probability is higher (“since this number is drawn particularly often”). By the way, these results stem from a didactic study of mathematics conducted by Efraim Fischbein and Ditza Schnarch in 1997. 

Slide 10

Once again: Random events do not have memory. This comes from an article from the magazine in the Süddeutsche Zeitung, a German daily newspaper. One reader has played the same numbers in a lottery syndicate for years. And sometimes the reader also uses these numbers on a Wednesday.

The problem: If the reader hits the big jackpot on a Wednesday, does he or she have to share the winnings with the lottery syndicate?

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The answer is brilliant, but especially this part is flawless from a mathematics perspective (even though a journalist answered and not a mathematician): “Wednesday – I promise you this and accept full responsibility for the following statement – couldn’t care less what numbers you bet on Saturday.”

That’s exactly right. I repeat: Random events really do not have memory.

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A die was rolled 100 times. In your opinion, which chart truly shows a result honestly determined by a die and thus a randomly generated result?

The chart on the right appears enticing, but actually the chart on the left is the result of really rolling a die. Perhaps you remember this talk about mathematics from the first episode.

I made up the chart on the right. Of course this doesn't mean that precisely these rolls couldn’t really occur. However, we tend to regard results that are closer to a sort of mean value as more probable. But this also doesn't matter to random events in principle.

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What we have just seen are misconceptions about repeated experiments.

People distinguish other problems in the identification of experiment with the same structure: The probability of three 5s when three dice are rolled at once or when one die is rolled three times is equal.

There are also typical misconceptions about combined experiments. Let’s look at an example: Person M wears lipstick every day. What is more likely? “Person M plays the trumpet” or “Person M is a woman who plays the trumpet”?

Yes, in this case “woman” is a constraint that is reasonable. However, it is a constraint as compared to simple trumpet playing, and so the probability of “person M plays the trumpet” is greater.

We’re set on the wrong track twice in this case. First, rather few people play the trumpet, and second, most people who use lipstick are likely women. The combined experiment is somewhat different from the sum of its parts.

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Exciting didactic research in mathematics is being conducted in this regard. And it shows that sometimes over time people learn correctly.

Let’s look at this exercise: Peter has flipped a coin three times and come up with heads each time. He flips the coin again. The probability of coming up with heads again is

1. less than
2. greater than
3. the same as

 the probability of coming up with tails.

The table shows that somewhat less than half of the children in grade 5 judge this correctly. By contrast, nearly all participants in grade 11 and at the university level who took part in this empirical study answered the question correctly.

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The results look a bit different for this exercise: Two dice are rolled at once. What is more likely?

1. a) Rolling a 5 and 6
   b) Rolling a pair of 6s
   c) Both are equally likely

In grade 5, 15% of the children give the correct answer, in grade 11, one quarter, and at the university level, hardly anyone. We don’t have to scrutinize the numbers in detail, but they show a clear trend that such an exercise is and remains difficult for many test subjects.

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In closing, let’s look at the animistic perception. In this case, children assume that beings with personal characteristics know the outcome of an experiment in advance (and poss. “give away”), influence it (e.g., on request), or help influence it in the sense of a teammate or opponent.

The question here is: Is there a being in the die that ensures that a 6 comes up?

It’s clear that as adults, we know that this is certainly not the case. But sometimes that would be very practical, right?

Slide 17

These were a few ideas about stochastics and related misconceptions. There are more, no doubt, but I hope you have gained initial insight. Thank you for your attention.