6 Drawing objects from an urn: Sometimes the order is important.

Slide 0

Welcome to this episode in which we talk again about math. We will focus on random experiments in which the order also counts for the result. Of course, it makes a difference whether a runner in a competition comes in first, second, or third place. However, we’re less interested in athletic success and more interested in the principle possibilities.

Slide 1

Let’s start with an example. ASCII, the American Standard Code of Information Interchange, is used to code information. It follows a simple logic: A byte exists. It consists of eight storage units that can be either switched on or switched off. If you set the one state to 0 and the other to 1, then there is the possibility to fill each of these eight places with 0 or 1. Accordingly, you can form

2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 = 2⁸ = 256

words with a length of 8 characters. The order is important in this case, because 0101010101 is different from 10101010.

This quantity is sufficient, for example, to code upper- and lowercase letters of the Latin alphabet, the digits from 0 to 9, and a couple of punctuation marks and control characters.

Slide 2

A soccer pool is also an example that the order can be important. In Germany, this is a bet where someone must predict the outcome of 13 matches, that is, whether the team mentioned first will win or lose or whether the teams will end in a tie. Accordingly, there are 3¹³ = 1,594,323 possibilities for filling out the betting slip. Each of the 13 matches can have each of the three possible outcomes.

Slide 3

Generally speaking, this is nothing more than drawing objects and replacing them while taking the order into account.

Example

Three distinguishable balls are in an urn. We draw a ball twice.

How many different possibilities exist if we draw a ball, replace it, and draw a second time? The order should play a role in the process.

Slide 4

Once again, the solution is immediately evident and very simple: There are 3 • 3 = 3² = 9 possibilities.

Slide 5

And if you wish, you can also use a couple more balls.

Eight distinguishable balls are in this urn. In how many ways can we draw five balls if we always replace them and we’re also interested in the order?

In this case as well, we just simply have to create an adequate formula and come up with

8 • 8 • 8 • 8 • 8 = 8⁵ = 32,768

possibilities for the different draws.        

Slide 6

Let’s look at the exercises again from the systematic perspective. We draw a ball, replace it, and we’re interested in the exact sequence. We thus draw balls with replacement while taking the order into account.

The basis is a set with n elements from which we draw a series of k elements. These are so-called k-tuples in which the individual positions can be filled repeatedly with the same elements. There are n^k possibilities for this.

Slide 7

Consider other applications as in this example: Hanna has a combination lock with five places. How many different combinations can she enter? How many different combinations are there if she uses only the digits 1, 2, and 3?

Slide 8

Let’s look again at the horse race. If you would like to know how many outcomes the race can have if 20 horses start and also cross the finish line, you can easily determine this. Twenty horses can possibly come in first place, 19 horses in second place, 18 horses in third place, 17 horses in fourth place, etc. Altogether you arrive at

20! = 20 • 19 • 18 • 17 • … • 3 • 2 • 1

possibilities. But who is interested in this only from the theoretical perspective?

In practical terms, it is usually far more interesting to know who comes in the first three places. How many possibilities are there for that?

Slide 9

Are you familiar with the “trifecta” in horse racing? This bet is all about selecting the three horses – for example, out of 20 horses at the start – that come in the first three places.

How do you calculate this? Well, you multiply, what else. Clearly there are

20 • 19 • 18 = 6,840

possibilities.

Slide 10

This exercise follows the same pattern: If ten people sit on four distinguishable chairs, there are 10 • 9 • 8 • 7 = 5,040 different possible ways to do this.

Slide 11

And now we’ll try again to consider the situation systematically: We draw a ball, do not replace it, but we’re interested in the exact sequence. This is about sampling without repetition (thus without replacement) while taking the order into account.

Here you see another example, this time with balls and an urn.

Eight different balls are in this urn. We draw five balls without replacing them. How many different possibilities are there?

Slide 12

Here is the solution: According to the general counting principle, there are eight possibilities for the first place, seven possibilities for the second place, six possibilities for the third place, five possibilities for the fourth place, and four possibilities for the fifth place. That comes to 8 • 7 • 6 • 5 • 4 different possibilities.

If we start from the permutations of eight elements, then we could also write 8! divided by 3! because the possibilities determined by the three remaining balls are omitted in this case.

Slide 13

And in general, the formula looks like this: The number of k-permutations of a set with n elements is n! divided by (n-k)!.

Slide 14

Look at more examples and solve the exercises:

• WAIT TO BE SEATED ... Twelve tables are open at a restaurant. Five couples arrive. How many ways are there to assign them to their own table?
• In the Olympic 100 meter dash, eight sprinters reach the final heat. How many ways are there theoretically to distribute the three medals?
• Three people board a bus with eight empty seats. In how many ways can they occupy the seats?

Slide 15

Once again: We use the urn model to describe the situations.

In an urn (or in a pouch; at any rate it must be opaque) there are balls that differ only in their color or numbering (in any case, not in their shape). Using this urn, we can simulate many of the common random experiments if the urn is filled with the right number of balls.

Slide 16

For example, let’s simulate the trifecta. We draw three balls in succession from an urn with 20 balls and take the order into account. This corresponds to the horse race on slide 9. And if we remove ten balls, then we can simulate the experiment with the chairs from slide 10. Urns are flexible; we can decide for ourselves how they are filled and how the objects are drawn.

Slide 17

And there are many possibilities for implementing an urn and the balls it contains, which should make its use in class simple. The urn can be a small bag, and the balls can be replaced with suitable blocks. 

Slide 18

Are you interested in applications? Think of your own exercises related to sampling without replacement while taking the order into account. Of course, they should be suitable for class.

Slide 19

That’s all for today. Thanks for being here and thank you for your interest.