Survey: Sales Approach In a sales effectiveness seminar, a...

Survey: Sales Approach In a sales effectiveness seminar, a group of sales representatives tried two approaches to selling a customer a new automobile: the aggressive approach and the passive approach. For 1160 customers, the following record was kept: $$\begin{array}{llcc} \hline & \text { Sale } & \text { No Sale } & \text { Row Total } \\ \hline \text { Aggressive } & 270 & 310 & 580 \\ \text { Passive } & 416 & 164 & 580 \\ \text { Column Total } & 686 & 474 & 1160 \\ \hline \end{array}$$ Suppose a customer is selected at random from the 1160 participating customers. Let us use the following notation for events: $A=$ aggressive approach, $P a=$ passive approach, $S=$ sale, $N=$ no sale. So, $P(A)$ is the probability that an aggressive approach was used, and so on. (a) Compute $P(S), P(S | A),$ and $P(S | P a).$ (b) Are the events $S=$ sale and $P a=$ passive approach independent? Explain. (c) Compute $P(A \text { and } S \text { ) and } P(P a \text { and } S).$ (d) Compute $P(N)$ and $P(N | A).$ (e) Are the events $N=$ no sale and $A=$ aggressive approach independent? Explain. (f) Compute $P(A \text { or } S).$

Survey: Sales Approach In a sales effectiveness seminar, a group of sales representatives tried two approaches to selling a customer a new automobile: the aggressive approach and the passive approach. For 1160 customers, the following record was kept:
$$\begin{array}{llcc}
\hline & \text { Sale } & \text { No Sale } & \text { Row Total } \\
\hline \text { Aggressive } & 270 & 310 & 580 \\
\text { Passive } & 416 & 164 & 580 \\
\text { Column Total } & 686 & 474 & 1160 \\
\hline
\end{array}$$
Suppose a customer is selected at random from the 1160 participating customers. Let us use the following notation for events: $A=$ aggressive approach, $P a=$ passive approach, $S=$ sale, $N=$ no sale. So, $P(A)$ is the probability that an aggressive approach was used, and so on.
(a) Compute $P(S), P(S | A),$ and $P(S | P a).$
(b) Are the events $S=$ sale and $P a=$ passive approach independent? Explain.
(c) Compute $P(A \text { and } S \text { ) and } P(P a \text { and } S).$
(d) Compute $P(N)$ and $P(N | A).$
(e) Are the events $N=$ no sale and $A=$ aggressive approach independent? Explain.
(f) Compute $P(A \text { or } S).$

Key Concepts

Joint Probability

Joint probability is the probability of two events occurring simultaneously and is represented as P(A and B). It is used to analyze the relationship between two events and is a key component in problems involving multiple events or outcomes.

Independence of Events

Independence of events occurs when the occurrence of one event does not influence the occurrence of another. Mathematically, two events A and B are independent if and only if P(A and B) = P(A) * P(B). Understanding independence is vital for correctly analyzing the relationship between different events.

Contingency Table Analysis

Contingency table analysis involves organizing data into a matrix format to display the frequency distribution of variables and facilitate the computation of probabilities, joint probabilities, and conditional probabilities. It is a useful tool for understanding relationships between categorical variables in statistics.

Probability

Probability is the study of quantifying the chance that an event will occur, typically defined as the ratio of the number of favorable outcomes to the total number of outcomes in a given sample space. It forms the foundation for all further analyses in probability theory and statistics.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has already occurred. It is calculated using the formula P(A|B) = P(A and B) / P(B), where P(B) > 0, and it is crucial in situations where events are interdependent.

Addition Rule (Probability of the Union of Events)

The addition rule is used to find the probability that at least one of two events occurs and is given by P(A or B) = P(A) + P(B) - P(A and B). This rule helps in preventing double counting when events have overlapping outcomes.

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