Probability distributions from empirical data | Probability & combinatorics

We're told that Jada owns a restaurant where customers can make their orders using an app. She decides to offer a discount on appetizers to attract more customers, and she's curious about the probability that a customer orders a large number of appetizers. Jada tracked how many appetizers were in each of the past 500 orders.

All right, so the number of appetizers: 40 out of the 500 ordered zero appetizers. And for example, 120 out of the 500 ordered three appetizers, and so on and so forth. Let x represent the number of appetizers in a random order. Based on these results, construct an approximate probability distribution of x. Pause this video and see if you can have a go at this before we do this together.

All right, so they're telling us an approximate probability distribution because we don't know the actual probability. We can't get into people's minds and figure out the probability that the neurons fire in exactly the right way to order appetizers. But what we can do is look at past results—empirical data, right? Over here to approximate the distribution.

So we can do is look at the last 500, and for each of the outcomes, think about what fraction of the last 500 had that outcome, and that will be our approximation. The outcomes here are: we could have zero appetizers, one, two, three, four, five, or six.

Now the approximate probability of zero appetizers is going to be 40 over 500, which is the same thing as 4 over 50, which is the same thing as 2 over 25. So I'll write 2/25 right over there. The probability of one appetizer, well, that's going to be 90 over 500, which is the same thing as 9 over 50. I think that's already in lowest terms.

Then, 160 over 500 is the same thing as 16 over 50, which is the same thing as 8 over 25, and we just keep going. 120 out of 500 is the same thing as 12 out of 50 or 6 out of 25—6 out of 25. And then, 50 out of 500, well, that's one out of every 10, so I'll just write it like that.

30 out of 500 is the same thing as 3 out of 50, so I'll just write it like that. And last but not least, 10 out of 500 is the same thing as 1 in 50. And we're done! We have just constructed an approximate probability distribution for our random variable x.