The accompanying table shows proportions of computer...

The accompanying table shows proportions of computer salespeople classified according to marital status and whether they left their jobs or stayed over a period of 1 year. $$ \begin{array}{lcc} \hline & {}{}{\text { Time on job }} \\ \text { Marital Status } & \geq \text { one year } & <\text { one year } \\ \hline \text { Married } & 0.64 & 0.13 \\ \text { Single } & 0.17 & 0.06 \\ \hline \end{array} $$ a. What is the probability that a randomly chosen salesperson was married? b. What is the probability that a randomly chosen salesperson left the job within the year? c. What is the probability that a randomly chosen single salesperson left the job within the year? d. What is the probability that a randomly chosen salesperson who stayed in the job over the year was married?

The accompanying table shows proportions of computer salespeople classified according to marital status and whether they left their jobs or stayed over a period of 1 year.
$$
\begin{array}{lcc}
\hline & {}{}{\text { Time on job }} \\
\text { Marital Status } & \geq \text { one year } & <\text { one year } \\
\hline \text { Married } & 0.64 & 0.13 \\
\text { Single } & 0.17 & 0.06 \\
\hline
\end{array}
$$
a. What is the probability that a randomly chosen salesperson was married?
b. What is the probability that a randomly chosen salesperson left the job within the year?
c. What is the probability that a randomly chosen single salesperson left the job within the year?
d. What is the probability that a randomly chosen salesperson who stayed in the job over the year was married?

Key Concepts

Contingency Table

A contingency table is a type of table in a matrix format that displays the frequency or proportion of observations for two or more categorical variables. It is used to analyze the relationship between the variables and to compute various probabilities by summing along rows, columns, or the table as a whole.

Marginal Probability

Marginal probability refers to the probability of a single event occurring without consideration of any other variable. It is obtained by summing the probabilities across the various levels of the other variables, typically along the margins of a contingency table.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is calculated using the ratio of the joint probability of both events to the probability of the conditioning event, providing insight into the dependencies between events.

Transcript

00:01 So we have a chart that's dealing with being married or single, and when we left a job, if someone left a job after they worked there for over a year, and that was 0 .64, 0 .17, those two add up to that column total is, what, 0 .81, and then left in under a year, that is 0 .13 and 0 .06.

00:26 And so this adds up to .19.

00:29 And if we add these two just to have our row totals and column totals having our frequencies, relative frequencies here, we have our marginal frequencies.

00:39 We have that's going to be 0 .77, and then these would add to 0 .23.

00:43 And notice these do add to 1, and these two do add to 1.

00:47 So our first question asks, what's the probability we randomly choose a person, and the person is married? and so that is the sum of these two.

00:57 It is the 0 .77.

01:00 What is the probability that you randomly chosen person leaves in less than a year? and that is going to be this column total of 0 .19.

01:12 Then on part c, we have what is the probability that given that a person left within a year that the person, oh, i'm sorry, i just said it the wrong way.

01:26 What is the probability that given that the person was single and that they left within a year? so again, the wording of that question we have to be careful of as to whether it's an and or a conditional.

01:41 And it says what is the probability randomly chosen single person left within a year? so we know the person was single and we want to know what's likelihood they left within a year.

01:53 So single is this 0 .23, and the part of that row total that left within a year is this 0 .06...

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