0:05

So, previously we talked about

Â understanding not only the expected level of variables, but

Â also the amount of variation that exists within those variables.

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

Â probability.

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

Â these concepts matter.

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

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

Â the probability foundations are clear.

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

Â the clearest way of being able to explain probability.

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

Â in marketing consulting contexts.

Â 0:51

The framework that I had laid out previously.

Â First we want to examine the relationships among variables whether those

Â are categorical variables, whether those are quantitative variables.

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

Â in those relationships.

Â Because uncertainty is caused by a couple of factors.

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

Â it could also be things that are beyond our control.

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

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

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

Â how popular products are going to be.

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

Â for product.

Â When we target customers with direct marketing.

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

Â our marketing materials.

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

Â to respond to those marketing materials.

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

Â our decisions.

Â 1:57

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

Â and we're trying to have a prediction for

Â how much revenue are we going to be bringing in.

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

Â how many customers we retain.

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

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

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

Â than we predicted.

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

Â than we expected.

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

Â net present value analysis.

Â From an inventories management standpoint and forecasting.

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

Â interruptions in the supply chain might come in.

Â If we're dealing with online advertising.

Â Online advertising is typically done in an auction format.

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

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

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

Â those key phrases.

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

Â 3:26

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

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

Â Of those people what fraction actually click on the ad?

Â Investment returns, you have the lifetime of products,

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

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

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

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

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

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

Â But, you might have problems with it.

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

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

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

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

Â average return or what's our average lifetime.

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

Â best guess.

Â 4:31

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

Â if I were able to collect millions and

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

Â Again, think of it as an average.

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

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

Â Well the probability gives us that long-run frequency.

Â How many outcomes do I actually observe meeting a criteria

Â divided by the total number of outcomes I observed.

Â 6:07

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

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

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

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

Â So how frequently do people buy Coca Cola and

Â buy shredded cheese, that's the joint probability.

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

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

Â That is what the upside down U is indicating.

Â The intersection of event A and event B happening.

Â 6:48

Another term that you may come across is conditional probability.

Â Given that an event has occurred.

Â Given that I am buying Coca Cola on this trip.

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

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

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

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

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

Â If we have an exhaustive set of possible outcomes,

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

Â All right.

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

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

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

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

Â That's going to add up to 100%.

Â 7:43

We might also talk about the complement, or

Â what's the probability of A not occurring.

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

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

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

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

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

Â versus the probability of it being a failure?

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

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

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

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

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

Â Just to give a visual representation of this,

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

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

Â B or A then what looks like a U.

Â And so how can this happen?

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

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

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

Â The likelihood of B occurring.

Â Sometimes when A happens, B also happens.

Â So, we're also double counting the probability

Â of both of these events happening together.

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

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

Â I've got the probability of B.

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

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

Â So that's the first sphere.

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

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

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

Â which is the joint probability of A and B.

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

Â And then what we're left with is.

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

Â All right so here just a couple of examples.

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

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

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

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

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

Â Could be late on your student loans.

Â Well, being late on any one of them.

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

Â 10:46

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

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

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

Â If A and B are independent of each other,

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

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

Â the product of B.

Â But what if there is a relationship between these?

Â 11:15

When A occurs, that changes the probability of B.

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

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

Â It can be phrased in terms of the probability of

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

Â It can also be phrased as the probability of B occurring

Â multiplied by the probability of A given B.

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

Â 11:54

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

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

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

Â costumer two default.

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

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

Â So probability of event A occurring and event B occurring.

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

Â So, either event A occurs or event B occurs.

Â