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In this lecture, I want to tell you about probability tables.

Why do we use tables?

So far, our calculations were always on rather small examples.

There was an A and a B, an intersection of A and B and so on.

Often, however, we want to look at probabilities in more complex examples and

we have many, many different probabilities.

We need to order them.

We need to do some accounting and get some structure on them, and

that's what probability tables are good for.

Let me show you a first example of data.

Here we have from a most recent year where complete data is available,

the number of foreign visitors to Switzerland.

Our friends in Australia or Suriname may forgive me.

I left you out, because your numbers weren't that large.

So let's divide the world into four continents, Africa,

the Americas, Asia and Europe.

Whether that's in Switzerland can stay overnight,

either in a hotel or in other destination and

other destination could be a caravan, could be a camping ground.

It could be a youth hostel.

It could be France.

It could be a private house or a private apartment that people rent.

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We see here in this particular year,

they were about 28.3 million visitors in total.

And so boredom,

we see a total of overnight stays by foreign visitors from the four continents.

Not surprisingly, most visitors came from Europe.

So, it's in Europe and

fewer people came from other continents where you effectively have to fly.

We see that about 17 million overnight stays were in hotels and

a bit more than 11 million were in these other domiciles.

Now I think you'll agree with me, this table is a mess.

We see these long numbers.

We have numbers between 50,000 and 28 million.

It's difficult to solve, look and understand this.

So this table of raw data, of counts,

we typically like to translate into proportions and

then you can also use them as probabilities using,

once again, concept number two, empirical probabilities.

Here in the bottom right-hand corner, this green shade, there's a reservoir.

100%.

How do I get this data?

If you look at the accompanying Excel spreadsheet,

you will see you just divide every number by the total.

28.3 million or the total divided by itself gives me this 1.000.

We see that 60% of all overnight stays were in hotels, 40% were other.

Those together, 60 plus 40, gives me the one.

At the bottom, we see the percentages for the four continents.

1.2% for Africa, 7 and a half for the Americas, 7.1 for Asia, 84.2 in Europe.

You add those numbers up and guess what?

We are back to 100%.

In the middle, we see now intersection probabilities.

For example, the upper left-hand corner, hotel and Africa 0.10.

So 1% of all overnight stays, whereby visitors from Africa's in hotels and

we have eight of these intersection probabilities.

Here now, a little bit of lingo from these probability tables,

the numbers in the margins of the table.

So, it's a very right column in the very bottom row are called marginal

probabilities.

Here for hotel, we would say, P of hotel is 0.6.

At the bottom, the P of Americas, 7.5%.

P of Asia, 7.1%.

So people in probability theory are not very creative or

imaginative, these are indeed the margins of the table and

that's where the name came from, marginal probabilities.

In the interior, in the middle, we have the intersection probabilities.

As I mentioned before, probability or hotel and Africa,

1% probability of other and Europe, 38.1%.

These intersection probabilities in the interior are called joint probabilities,

because we're looking at joint events of hotel and Africa joining and

both happening together.

That's the motivation for the choice of the name.

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So, that's what it means to be completely exhaustive.

Mutually exclusive means, either one or the other happens.

So don't tell me, I smoked, but didn't inhale.

That's nonsense, either you smoked or you did not smoke.

So that's an empty intersection and everything is covered and

then we have the part joined probabilities,

these intersection probabilities in the interior.

The probabilities in the margins, the margin probabilities

are then the sums across the rows or across the columns.

And so here now, let me show you this.

If you look at the first column, the Africa column,

1% plus 0.2% equals 1.2% in the total.

If you look at the row of other,

we have 0.002 plus 0.011 plus

0.007 plus 0.381, the total is 40%.

So, the margins is the sum of the interior.

Now, probability tables are very helpful to show us that marginal or

total probability of an event.

Probability of hotels or probability of Asia and

it's great at showing us these intersection probabilities.

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What is the probability that an overnight stay in

Alpha is from a person visiting from Europe.

What do we do?

Remember the conditional probability definition.

Take the intersection probability and

divide it by the probability of the event that occurred.

So here, the probability of Europe given other.

Take the intersection probability, that's the probability in the middle,

in the interior of the table and divide it by the appropriate marginal probability.

Here we learn more than 95% of all over night stays in other,

in caravans, in camping grounds, in vacation houses or apartments.

In youth hostels is from European visitors.

And here in the table, notice what we did.

We took the 0.381, the interior probability, Europe and

other and divided by the probability of other, 0.4.

So, you can just take the element from the interior of the table divided by

the marginal probability.

And voila, there's your conditional probability.

We can also go in the other direction, other given Europe.

So, what is the probability that a European visitor

will stay an overnight in other?

So now Europe is given,

probability of other given Europe is what we're looking for.

And now we take the same intersection probability, but

we're dividing by the marginal probability in the bottom row, the totals.

So, 0.38 divided by 0.842 and

that gives me 45%.

To sum up, why do we use probability tables?

Things can get very quickly, very ugly if you have many different events.

So, we need to learn to structure the presentation of the probabilities.

Probability tables are great way to do this, that's why we use them.

In the probability tables, we see marginal probabilities and joint probabilities.

And so, that's why we needed you to talk about these concepts.