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

Â Why do we use tables?

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

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

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

Â we have many, many different probabilities.

Â We need to order them.

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

Â that's what probability tables are good for.

Â Let me show you a first example of data.

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

Â the number of foreign visitors to Switzerland.

Â Our friends in Australia or Suriname may forgive me.

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

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

Â the Americas, Asia and Europe.

Â Whether that's in Switzerland can stay overnight,

Â either in a hotel or in other destination and

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

Â It could be a youth hostel.

Â It could be France.

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

Â 1:31

We see here in this particular year,

Â they were about 28.3 million visitors in total.

Â And so boredom,

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

Â Not surprisingly, most visitors came from Europe.

Â So, it's in Europe and

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

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

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

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

Â We see these long numbers.

Â We have numbers between 50,000 and 28 million.

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

Â So this table of raw data, of counts,

Â we typically like to translate into proportions and

Â then you can also use them as probabilities using,

Â once again, concept number two, empirical probabilities.

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

Â 100%.

Â How do I get this data?

Â If you look at the accompanying Excel spreadsheet,

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

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

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

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

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

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

Â You add those numbers up and guess what?

Â We are back to 100%.

Â In the middle, we see now intersection probabilities.

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

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

Â we have eight of these intersection probabilities.

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

Â the numbers in the margins of the table.

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

Â probabilities.

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

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

Â P of Asia, 7.1%.

Â So people in probability theory are not very creative or

Â imaginative, these are indeed the margins of the table and

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

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

Â As I mentioned before, probability or hotel and Africa,

Â 1% probability of other and Europe, 38.1%.

Â These intersection probabilities in the interior are called joint probabilities,

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

Â both happening together.

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

Â 6:02

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

Â Mutually exclusive means, either one or the other happens.

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

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

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

Â then we have the part joined probabilities,

Â these intersection probabilities in the interior.

Â The probabilities in the margins, the margin probabilities

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

Â And so here now, let me show you this.

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

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

Â If you look at the row of other,

Â we have 0.002 plus 0.011 plus

Â 0.007 plus 0.381, the total is 40%.

Â So, the margins is the sum of the interior.

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

Â total probability of an event.

Â Probability of hotels or probability of Asia and

Â it's great at showing us these intersection probabilities.

Â 7:39

What is the probability that an overnight stay in

Â Alpha is from a person visiting from Europe.

Â What do we do?

Â Remember the conditional probability definition.

Â Take the intersection probability and

Â divide it by the probability of the event that occurred.

Â So here, the probability of Europe given other.

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

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

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

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

Â In youth hostels is from European visitors.

Â And here in the table, notice what we did.

Â We took the 0.381, the interior probability, Europe and

Â other and divided by the probability of other, 0.4.

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

Â the marginal probability.

Â And voila, there's your conditional probability.

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

Â So, what is the probability that a European visitor

Â will stay an overnight in other?

Â So now Europe is given,

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

Â And now we take the same intersection probability, but

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

Â So, 0.38 divided by 0.842 and

Â that gives me 45%.

Â To sum up, why do we use probability tables?

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

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

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

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

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

Â