A survey carried out for a supermarket classified customers...

A survey carried out for a supermarket classified customers according to whether their visits to the store are frequent or infrequent and whether they often, sometimes, or never purchase generic products. The accompanying table gives the proportions of people surveyed in each of the six joint classifications. $$ \begin{array}{lccc} \hline & & {\text { Purchase of Generic }} \\ \text { Frequency of } & & \text { Products } & \\ \text { Visit } & \text { Often } & \text { Sometimes } & \text { Never } \\ \hline \text { Frequent } & 0.12 & 0.48 & 0.19 \\ \text { Infrequent } & 0.07 & 0.06 & 0.08 \\ \hline \end{array} $$a. What is the probability that a customer both is a frequent shopper and often purchases generic products? b. What is the probability that a customer who never buys generic products visits the store frequently? c. Are the events "never buys generic products" and "visits the store frequently" independent? d. What is the probability that a customer who infrequently visits the store often buys generic products? e. Are the events "often buys generic products" and "visits the store infrequently" independent? f. What is the probability that a customer frequently visits the store? g. What is the probability that a customer never buys generic products? h. What is the probability that a customer either frequently visits the store or never buys generic products or both?

A survey carried out for a supermarket classified customers according to whether their visits to the store are frequent or infrequent and whether they often, sometimes, or never purchase generic products. The accompanying table gives the proportions of people surveyed in each of the six joint classifications.
$$
\begin{array}{lccc}
\hline & & {\text { Purchase of Generic }} \\
\text { Frequency of } & & \text { Products } & \\
\text { Visit } & \text { Often } & \text { Sometimes } & \text { Never } \\
\hline \text { Frequent } & 0.12 & 0.48 & 0.19 \\
\text { Infrequent } & 0.07 & 0.06 & 0.08 \\
\hline
\end{array}
$$a. What is the probability that a customer both is a frequent shopper and often purchases generic products?
b. What is the probability that a customer who never buys generic products visits the store frequently?
c. Are the events "never buys generic products" and "visits the store frequently" independent?
d. What is the probability that a customer who infrequently visits the store often buys generic products?
e. Are the events "often buys generic products" and "visits the store infrequently" independent?
f. What is the probability that a customer frequently visits the store?
g. What is the probability that a customer never buys generic products?
h. What is the probability that a customer either frequently visits the store or never buys generic products or both?

Key Concepts

Marginal Probability

Marginal probability is the overall probability of an event occurring, regardless of the intersection with another event. It is derived by summing the probabilities of all events in a row or column of a contingency table, reflecting the total proportion for one variable.

Union of Events

The union of events involves calculating the probability that at least one of several events occurs. This is determined by adding the individual probabilities and subtracting the probability of their overlap to avoid double counting, reflecting the total likelihood of one or more events happening.

Independence

Independence between two events means that the occurrence of one event does not affect the probability of the other. Formally, two events are independent if the joint probability equals the product of their respective marginal probabilities.

Joint Probability

Joint probability refers to the probability of two events occurring simultaneously. In the context of a two-way classification, it considers the proportion or likelihood that both criteria or conditions are met at the same time.

Conditional Probability

Conditional probability is the probability that one event occurs given that another event has already occurred. It is calculated by taking the joint probability of both events and dividing it by the marginal probability of the conditioning event, helping to understand the dependency between events.

Transcript

00:01 So we have a big table and we have purchasing generic products and that happens often sometimes or never.

00:09 I buy generic if i can.

00:11 They're cheaper.

00:12 And if they're just as good, then why not? anyway, so we have our grid looks like this and i'm going to do these row totals and column totals.

00:24 I do that to begin with on a frequency distribution like this.

00:30 Doesn't take very long to do and i can sometimes catch that i wrote a number down wrong when they don't add up to the way they should because these three need to add up to 1.

00:39 These add up to 0 .79 and these add up to 0 .21 and these two should add up to that.

00:45 So let's start with the questions.

00:47 We want to find what is the probability that someone is a frequent shopper and they often buy generic and that is going to be 0 .12.

01:02 What is the probability that someone never buys generic and shops frequently? and never buys and shops frequently, that's that 0 .19.

01:16 Then part c .s are these two, this and never buying generic and shopping frequently, are they independent? and if they're independent, the probability of both happening, this and this, which is 0 .19, should equal the probability of never times the probability of frequent.

01:41 And so probability of never is 0 .27.

01:46 Probability of being a frequent shopper is 0 .79.

01:50 And let's see what we get.

01:51 0 .27 times .79 comes out to be 0 .2133.

02:01 And notice that these two are not equal.

02:05 So they are not independent.

02:08 This is how often they happen together and they should happen together two and a third percent more often than that if they were independent.

02:17 So there's a little bit lesser chance than what you would expect.

02:20 So they're not independent.

02:22 So let's move on.

02:24 Letter d, we want to find what is the probability that given someone is an infrequent shopper, so we only care about this row, what is the likelihood that that person is often buying the generic product? so our probability is we're only concerned about that 21%, and how many of those are in the option category.

02:52 0 .07.

02:54 And so that looks like that's one -third to me...

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