1 Multistage random experiments

Slide 0

I hope you’re all having a wonderful day. Once again we want to talk about math, and today we’ll focus on multistage random experiments. We will look at examples and consider suitable representations for their results. In the process, we will also show how probabilities can be usefully assigned to the results.

With this episode we are starting our “stochastics for professionals” category, but don’t worry – you’ll be able to understand it without great difficulty.

Slide 1

Today we will talk about multistage random experiments. To prepare you for this topic, we will first explain what a single-stage random experiment is.

It’s quite clear. We speak of this if we conduct a random experiment exactly one time.

Here are some examples. For instance, you might be interested in the popularity of math in a school class. This results in one question and one answer. But you might want to know whether the children in a class are in a sports club or not. Simple question and simple answer, because here the children can decide only between “yes” and “no.”

Flipping a coin once or rolling a die once is also a single-stage random experiment.

In the same way, you can also look at a medical test as this type of random experiment. In this case, the answer does not necessarily have to be reduced to only “positive” and “negative.” For example, for blood types there are four possibilities: O, A, B, and AB. And for blood pressure, we cannot really list all the possibilities. But this example might not be so exciting for school.

Slide 2

So someone is interested in the popularity of math in a class and asks the simple question “Do you like the subject?” with the possible answers “yes” or “no.”

This is what it looks like in the tree diagram: Super simple. There are one node and two branches.

Slide 3

Now what is a multistage random experiment?

Also very easy. It involves multiple random experiments that are normally conducted one after the other. You can repeat the same random experiment, but you can also combine different random experiments in this way.

Let’s look at examples.

For instance, you might be interested in the popularity of English and math in a school class. Then you would carry out two polls in succession.

Or you would like to know whether the children in a class play an instrument and/or whether they are in a sports club. Again, that’s two polls or at least two different points on a questionnaire.

The old, familiar random experiments of flipping a coin or rolling a die can be carried out multiple times.

Similarly, you can also carry out multiple medical tests, for instance, to confirm a diagnosis. Or to obtain a comprehensive picture of a person’s state of health.

Slide 4

Let’s take a closer look at one of these experiments.

We want to take a closer look at the examples and begin with the popularity of the subjects English and math in a class. This brief survey is actually sufficient as a start to understanding basic aspects.

We ask the simple question “Do you like the subject?” with the possible answers “yes” or “no.” Then there are four possible answers. A person can like exactly one, or none, or both of the two subjects. In this case as well, you can represent the possibilities in a tree diagram.

“Person likes the subject English” and “Person does not like the subject English” appear in the nodes as do “Person likes the subject math” and “Person does not like the subject math.”

Slide 5

You could just as easily use a 2x2 table to represent the results. It is also suitable for representing the results in a way that’s very easy to read. Let’s look at this using a practical example.

Slide 6

Let’s assume the questions were asked in a class with 28 students. Then the 2x2 table could look like this when it is completely filled in:

Of the 28 students, 14 like both math and English, 5 like only math, 6 like only English, and 3 like neither one nor the other subject. Then you can add up the numbers in the rows and columns and obtain individual statements in this way. For instance, 19 students like the subject math – regardless of what they think of the subject English. Look at the individual numbers and interpret them for yourself. But one thing is certain: The number 28 must appear at the lower right because there was one answer for each of the students in the class.

Slide 7

We have entered the absolute numbers in the 2x2 table, but normally the relative values are what’s interesting. These values can be easily determined by dividing the respective values by 28. Then the 2x2 table looks as shown here.

The sum of the two values in the bottom row and in the last column to the right must obviously be 1.

Slide 8

Let’s look at another example. We flip a coin twice and want to know whether it comes up heads or tails.

This experiment has four possible outcomes:

• You come up with “heads” both times.
• You come up with “tails” both times.
• You flip “heads” first and then “tails.”
• You flip “tails” first and then “heads.”

Slide 9

If you conduct this random experiment multiple times, you can enter the results in a 2x2 table in this case as well.

The results of 100 experiments are entered here, and this number can be seen very clearly in the lower right cell.

Slide 10

Obviously, you can also turn these absolute numbers into relative frequencies. This time it is even easier because dividing by 100 is really straightforward.

Slide 11

This random experiment is also two-stage. We have looked at it before. We roll a die twice and want to know whether a 6 is rolled at least once.

This experiment also has four possible outcomes:

• You roll a 6 both times.
• You do not roll a 6 either time.
• It works only on the first roll.
• It works only on the second roll.

Slide 12

We can use a tree diagram here that shows the various possible outcomes of the experiment very clearly.

Slide 13

What if it gets a little more complex? Let’s look at an example.

We roll a die three times and want to know whether at least one 6 is rolled. Clearly, these are the general results:

• You roll a 6 every time.
• You do not roll a 6 any of the times.
• It works only on the first roll.
• It works only on the second roll.
• It works only on the third roll.
• It works on the first and second rolls.
• It works on the first and third rolls.
• It works on the second and third rolls.

Slide 14

In this case, which representations make sense for the possible results?

Slide 15

It’s very practical that you can always use a tree diagram. It’s no surprise that the three-stage random experiment has one more level than the two-stage random experiment.

Slide 16

However, multiple stages do not necessarily mean that another random experiment is conducted for each prior result.

For instance, when someone is tested for COVID-19, usually an antigen test is done first, and a PCR test is not done unless the result of the antigen test is positive. You can easily represent this in a tree diagram. It has the necessary nodes precisely where a continuation is possible. A 2x2 table does not make sense here.

Slide 17

Let’s briefly summarize what we have talked about today.

We have looked at various multistage random experiments. These experiments involve conducting simple random experiments, or single-stage random experiments, in succession.

Today our primary goal was to understand the term and suitable representations.

We now know that it makes sense and is possible to represent the results in a tree diagram for all experiments. However, a 2x2 table is also possible only for two-stage random experiments – and also only if the two experiments are independent of each other.

Slide 18

That’s all for today. Thank you for your interest. Until next time!