Subscribers to a local newspaper were asked whether they...

Subscribers to a local newspaper were asked whether they regularly, occasionally, or never read the business section and also whether they had traded common stocks (or shares in a mutual fund) over the last year. The table shown here indicates the proportions of subscribers in six joint classifications. $$ \begin{array}{lccc} \hline {}{}{\begin{array}{l} \text { Traded } \\ \text { Stocks } \end{array}} & {}{}{\text { Read Business Section }} \\ & \text { Regularly } & \text { Occasionally } & \text { Never } \\ \hline \text { Yes } & 0.18 & 0.10 & 0.04 \\ \text { No } & 0.16 & 0.31 & 0.21 \\ \hline \end{array} $$ a. What is the probability that a randomly chosen subscriber never reads the business section? b. What is the probability that a randomly chosen subscriber has traded stocks over the last year? c. What is the probability that a subscriber who never reads the business section has traded stocks over the last year? d. What is the probability that a subscriber who traded stocks over the last year never reads the business section? e. What is the probability that a subscriber who does not regularly read the business section traded stocks over the last year?

Subscribers to a local newspaper were asked whether they regularly, occasionally, or never read the business section and also whether they had traded common stocks (or shares in a mutual fund) over the last year. The table shown here indicates the proportions of subscribers in six joint classifications.
$$
\begin{array}{lccc}
\hline {}{}{\begin{array}{l}
\text { Traded } \\
\text { Stocks }
\end{array}} & {}{}{\text { Read Business Section }} \\
& \text { Regularly } & \text { Occasionally } & \text { Never } \\
\hline \text { Yes } & 0.18 & 0.10 & 0.04 \\
\text { No } & 0.16 & 0.31 & 0.21 \\
\hline
\end{array}
$$
a. What is the probability that a randomly chosen subscriber never reads the business section?
b. What is the probability that a randomly chosen subscriber has traded stocks over the last year?
c. What is the probability that a subscriber who never reads the business section has traded stocks over the last year?
d. What is the probability that a subscriber who traded stocks over the last year never reads the business section?
e. What is the probability that a subscriber who does not regularly read the business section traded stocks over the last year?

Key Concepts

Joint Probability Distribution

A joint probability distribution displays the probability of two events occurring together and is typically presented in a table format where each cell shows the probability of the intersection of specific outcomes from each category. This allows us to analyze the relationship between the two events simultaneously.

Marginal Probability

Marginal probability refers to the probability of an event occurring without consideration of any other variables. It is obtained by summing the joint probabilities over the unwanted variable, providing a single probability for one variable across all categories of the other variable.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has occurred. It is computed by dividing the joint probability of both events by the marginal probability of the given condition, enabling partial insights into the relationship of the events.

Probability Tables

Probability tables organize data into rows and columns that represent different events or outcomes, making it easier to compute joint, marginal, and conditional probabilities. They serve as a visual tool to summarize and analyze complex probabilistic relationships.

Transcript

00:01 So we have the following distribution.

00:03 This is trading, and this is yes and no.

00:07 And this is reading the business section, and that happens regularly, occasionally, or never.

00:14 And so let's fill in this grid.

00:17 0 .18, 0 .1.

00:20 I don't know if we need that total, but we'll just add them.

00:22 0 .1, 0 .31, and again, this would be 0 .41, and never was 0 .04, and this is 0 .21.

00:29 And adding those would be 0 .25.

00:31 And by the way, if we add all of these together, we do add up to 1.

00:36 And let's see, this is 0 .28.

00:39 So that would add up to 0 .32.

00:41 I like to do these marginal, add up these marginal totals.

00:45 It's helpful.

00:46 So that's 0 .47 and 0 .47, and that comes up to be 0 .68.

00:51 And notice that these two add up to 1.

00:53 So let's start answering the questions.

00:54 So we have in the first one, what's the probability you randomly choose a person, and that person has never traded, or excuse me, had never read the business section, and that would be 0 .25.

01:09 Part b, what is the probability that someone has traded? and so that's going to be this row, which is 0 .32.

01:21 Part c, what is the probability, given that someone has not read the business section that they have traded? and so we know that this is, we are only dealing with this column.

01:36 And this column, you know, technically this is the probability of y and n divided by the probability of n...

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