0:05

So, previously we talked about

understanding not only the expected level of variables, but

also the amount of variation that exists within those variables.

And that's what really the focus if going to be in this session on randomness and

probability.

So the plan is let's make sure we're on the same page as far as why

these concepts matter.

And why they matter from a specifically from a marketing standpoint.

We'll go into some background just to make sure that

the probability foundations are clear.

I'm going to pull a couple of examples from casino games because that seems to be

the clearest way of being able to explain probability.

And then we're going to relate that back to decision making

in marketing consulting contexts.

0:51

The framework that I had laid out previously.

First we want to examine the relationships among variables whether those

are categorical variables, whether those are quantitative variables.

And we want to have a good understanding of how much uncertainty there is

in those relationships.

Because uncertainty is caused by a couple of factors.

It could be things that we don't know about but

it could also be things that are beyond our control.

So, different sources of uncertainty we want to factor that into

the decision making that we have and tie that to our evaluation model.

For example, the level of demand, we might have a best guess for

how popular products are going to be.

But we don't know with 100% certainty what the level of demand is going to be

for product.

When we target customers with direct marketing.

We've got the best guess as far as which customers are going to be receptive to

our marketing materials.

But we don't know with certainty which of those customers are actually going

to respond to those marketing materials.

So those are a couple of the places where we can see uncertainty affecting

our decisions.

1:57

To give you a few other contexts, maybe it's in the context of customer evaluation

and we're trying to have a prediction for

how much revenue are we going to be bringing in.

Well that's going to depend on how many customers we sign up and

how many customers we retain.

And when we're making these forecasts, especially when we're making forecasts in

the long run we don't know how many customers we're going to keep.

Maybe we're going to have some months when a lot more customers drop out

than we predicted.

Maybe we're going to have some months where retention is better

than we expected.

So we've gotta deal with uncertainty when we're doing customer evaluation and

net present value analysis.

From an inventories management standpoint and forecasting.

Again we don't know what demand is going to look like we don't know where

interruptions in the supply chain might come in.

If we're dealing with online advertising.

Online advertising is typically done in an auction format.

Whoever is willing to pay the most for a particular key phrase when we're dealing

with search advertising well they're the ones who are going to get it.

Which means it's all dependent on how much your competitors are going to pay for

those key phrases.

So your competitors' reactions may have an impact on your success in that context.

3:26

the number of hits on a website on a given day.

The conversion rates, so the number of people exposed to an ad.

Of those people what fraction actually click on the ad?

Investment returns, you have the lifetime of products,

these are all just context where we have a best guess.

But even though that's our best guess it doesn't mean that that

is exactly the value that we're going to get every single time.

You buy a new car, you buy a new phone.

And how long until you have to take it in for repairs?

Well hopefully it's a couple years, hopefully you don't have any problems.

But, you might have problems with it.

You might have to take that phone in two or three times for service.

You might have bought a lemon off the used car lot, and

now you gotta put more money into the repairs than you had anticipated.

So, that's what we need, not just the understanding of what's our

average return or what's our average lifetime.

We also need to understand how much uncertainty is there around that

best guess.

4:31

And what the probability refers to is if I were to run or

if I were able to collect millions and

millions of observations the probability is telling us the long-run frequency.

Again, think of it as an average.

But how likely is it that we are going to observe different values?

Or how likely is it that we're going to observe different outcomes?

Well the probability gives us that long-run frequency.

How many outcomes do I actually observe meeting a criteria

divided by the total number of outcomes I observed.

6:07

I have an increased tendency to buy Coca Cola when I buy shredded cheese.

Then these events are going to be dependent on each other.

If they're unrelated then going to be said to be independent of each other.

When we're interested in calculating the likelihood of two events happening.

So how frequently do people buy Coca Cola and

buy shredded cheese, that's the joint probability.

The notation that we're going to use for the joint probability,

it's going to be a probability of A and B occurring, or the intersection of A and B.

That is what the upside down U is indicating.

The intersection of event A and event B happening.

6:48

Another term that you may come across is conditional probability.

Given that an event has occurred.

Given that I am buying Coca Cola on this trip.

How likely is it that I'm going to buy shredded cheese?

The notation we're going to use there, B with a vertical slash and

then A, or being pronounced as the probability of event B given event A.

Couple of rules, and this may be a little bit of a refresher for you.

For any event A probability is going to fall somewhere between zero and one.

If we have an exhaustive set of possible outcomes,

the sum of those probabilities has to add up to one.

All right.

So if I only have two brands in the grocery store for soda,

I've got Coca Cola and I've got Pepsi.

What's the probability of buying Coca Cola plus the probability of buying Pepsi.

And I've said I'm buying soda on this trip.

That's going to add up to 100%.

7:43

We might also talk about the complement, or

what's the probability of A not occurring.

Well, the notation for A not occurring would be a probability of A with

a superscript c, can be calculated as one minus the probability of A.

So if we think about this as what's the probability that somebody submits

a complaint versus not submitting a complaint after bad customer service?

What's the probability of a product being successful in one launch

versus the probability of it being a failure?

These are examples where they're the complements of each other.

Either you choose to submit a complaint or you do not.

Either the product is a success or it's deemed a failure.

Well since they're the flip side of each other and the probabilities have to add up

to one, that's where we get one minus the probability being the complement.

Just to give a visual representation of this,

one of the rules of probability is the probability of A or B occurring.

And A or B, the notation for that, is the union of A and

B or A then what looks like a U.

And so how can this happen?

Either event A can occur, or event B can occur.

And that's what these first two pieces are indicated for us in this equation.

But then if you think about the likelihood of A occurring.

The likelihood of B occurring.

Sometimes when A happens, B also happens.

So, we're also double counting the probability

of both of these events happening together.

And so we actually have to subtract out the joint probability.

So what this looks like visually is I've got the probability of A,

I've got the probability of B.

And the overlap between those circles, that's the probability of A and B.

So what we want to calculate is how likely it is that A occurs?

So that's the first sphere.

How likely B occurs, so that's our second sphere.

And that's the probability of A plus the probability B.

But what we've ended up doing is double counting this shaded region,

which is the joint probability of A and B.

So that's why we've got to subtract that out.

And then what we're left with is.

The overall shaded area that's the probability of A or B occurring.

All right so here just a couple of examples.

You buy a particular brand at least once on your last two shopping trips.

So either you bought it on the first trip or you bought it on the second trip.

You also could have bought it, on both of those trips.

You make a late, another example, you're late paying at least one of your bills.

So, could be late on your mortgage, could be late on your car loan.

Could be late on your student loans.

Well, being late on any one of them.

So we're just, all that we're interested is that you're late on at least one bill.

10:46

So that's looking at addition, looking at the probability of A or B.

We might also look at two events happening together, probability of A and B.

And so that's where the multiplication rule comes into play.

If A and B are independent of each other,

the joint probability, the probability of A and B occurring.

That's just going to be given by the product of A multiplied by

the product of B.

But what if there is a relationship between these?

11:15

When A occurs, that changes the probability of B.

Or when B occurs, that changes the probability of A.

If you're looking for a more general multiplication rule.

It can be phrased in terms of the probability of

A occurring multiplied by the probability of event B given that A has occurred.

It can also be phrased as the probability of B occurring

multiplied by the probability of A given B.

Both of these are going to be equivalent to each other.

11:54

So, if we're looking at events jointly occurring, being late on multiple bills,

being late on two bills, or custom for a loan provider.

The likelihood of multiple customers defaulted, well customer one default and

costumer two default.

And if you think about the multiplication rule of being about joint events, really

the tip off to be using the multiplication rule is the phrase and, right.

So probability of event A occurring and event B occurring.

Whereas the addition rule, they keyword there is going to be or.

So, either event A occurs or event B occurs.