Now the expected value of people showing up is

still nowhere close to 200, tt's now 190.65.

Notice the expected revenue went up.

How did I calculate those?

Now let's go deep into the spreadsheet here.

If we sell 205 tickets,

then in the worst case 205 people show up.

My revenue, however, is only 60,000.

Why, because only 200 people will actually leave with us, 200 times 300.

Then I have the over cost of 2,500.

That means five people get over cost of 500, five people get 500 each,

2500, in addition to the 300 that I don't get,

makes revenue minus cost 57,500.

Now how likely is it that that happens?

Here, look at this probability,

it's a binormal distribution of having exactly 200 people show up.

If I sell 205 people show up, if I sell 205

tickets the probability of 93% showing up, that's essentially 0.

This worst case scenario just doesn't happen.

Let's look at now this case, 202 people show up.

Again the revenue's the same, 60,000, I can only take 60,000,

make only that because I can take only 200 with me,

2 people I have to bump, that's 500 in overage cost.

So I make net only 59,000, not 60,000, only 59,000.

What's the probability of that happening?

Still very tiny, and again I use a binomial distribution.

202 people show up, if I sell 205 tickets, but for

probability of showing up of 93%, that's one minus the 7%.

And so, now I have all the possible revenue minus cost.

Here I have all the possible probabilities, and

then I can calculate the expected value, which I do here in the column J.

I add up all these numbers.

Now, I can tell you the truth,

I don't actually do this for all the possible numbers.

Why, because at some point they're all zero, the probabilities are so tiny.

And here, I get the round number to two digits, 57,193.

Now, let's play with this, let's say we sell five more tickets, 210.

What happens to revenue minus cost?

It's still goes up.

Let's try 214, it still goes up!

Now let's go crazy, 230, not a good idea, it goes way down.

Why, because, suddenly it's very likely that we have to bump many people.

For example, if we sell now 230 tickets,

the probability of say 214 showing up is 10%.

That's rather high, so clearly it's not a good idea to sell 230 tickets.

Now how can you find the optimal number?

Here with some trial and error, you can play around with this number, and

it's around 214, 215 is the optimal number.

Now for those of you who know a little bit more Excel, if you have access to

the Excel server, I have hooked up an Excel server to this Excel sheet,

and with this server we can actually find the optimal solution here.

However that requires some optimization, which is not part of this class.

But for any people who have played around with the server before,

you may find that interesting.

Now let me summarize what we have seen here in the slides.

So, and move back to the slides.