Question

A salesperson figures that the probability of her making a sale during the first contact with a client is .4 but improves to .55 on the second contact if the client did not buy during the first contact. Suppose this salesperson makes one and only one callback to any client. If she contacts a client, calculate the probabilities for these events: a. The client will buy. b. The client will not buy.

Short answer

Answer: The probability that a client will buy is 0.73, and the probability that a client will not buy is 0.27.

Step-by-step solution

Determine the probability of making a sale on the first contact (P(S1))

The probability of a sale occurring during the first contact is given in the problem as 0.4. So, P(S1) = 0.4.

Determine the probability of NOT making a sale on the first contact (P(NS1))

The probability of not making a sale on the first contact is the complement of making a sale on the first contact. Hence, P(NS1) = 1 - P(S1) = 1 - 0.4 = 0.6.

Determine the probability of making a sale on the callback, given no sale on the first contact (P(S2|NS1))

The probability of making a sale on the callback, given that the client did not buy during the first contact, is given as 0.55. So, P(S2|NS1) = 0.55.

Calculate the probability of making a sale (P(S1 U S2|NS1)) using the law of total probability

We can now use the law of total probability to find the probability of making a sale, either on the first contact or on the callback. The formula is: P(S1 U S2|NS1) = P(S1) + P(S2|NS1) * P(NS1) = 0.4 + 0.55 * 0.6 = 0.4 + 0.33 = 0.73

Calculate the probability of not making a sale (P(NS1 ∩ NS2|NS1))

Since the probabilities of making a sale and not making a sale are complementary events, we can calculate the probability of not making a sale as: P(NS1 ∩ NS2|NS1)= 1 - P(S1 U S2|NS1) = 1 - 0.73 = 0.27 Now we have the probabilities for the two events: a. The probability that a client will buy is 0.73. b. The probability that a client will not buy is 0.27.

Key concepts

Law of Total Probability

The Law of Total Probability is a fundamental principle in probability theory that allows us to determine the total probability of an event by considering various potential scenarios or paths to that event.
This law is especially useful when dealing with events that can happen in multiple ways. In this scenario with our salesperson, we want to calculate the probability of making a sale. Let's break this down:

• First, we have the probability of making a sale on the first contact, which is 0.4 (or 40%).
• Then, we consider the scenario where no sale is made on the first contact, but a sale is achieved in the callback.

The probability of a sale happening in either of these ways can be put together as follows:The probability of making a sale on the first contact plus the probability of making a sale on the second contact given there was no sale initially. The calculation is: \[ P(S1 \cup S2|NS1) = P(S1) + P(S2|NS1) \times P(NS1) \]In formula terms, this is: \[ 0.4 + 0.55 \times 0.6 = 0.73 \]This gives us a probability of 0.73, meaning there is a 73% chance of making a sale, either on the first contact or on the callback. This comprehensive approach shines as the heart of the Law of Total Probability, ensuring every possible path is considered.

Complementary Events

Complementary events are events where the occurrence of one event means the other cannot happen. In simple terms, if something happens, its complement surely does not happen.
They are two sides of the same coin — both events together cover all possible outcomes in context.Looking at our salesperson's situation:- If the first event is making a sale, the complement is not making a sale.- Similarly, the complement of a successful callback sale (given no sale at first) is no sale on the callback.The probability of an event plus the probability of its complement always sums up to 1. This concept was crucial in Step 5 of our original solution. After calculating the chance of making a sale altogether, which was 0.73, its complement directly gives us the non-sale probability:- Non-sale probability= 1 - Probability of a sale - Thus, \[ P(NS1 \cap NS2|NS1)= 1 - 0.73 = 0.27 \]This calculation tells us there's a 27% chance of not making a sale at all, either on the initial contact or the follow-up.

Conditional Probability

Conditional Probability is the likelihood of an event occurring, given that another event has already occurred. It helps in narrowing down the probability based on some prior information.
For instance, knowing that the weather forecast predicts rain might change how you assess the chance of having an outdoor event.In our salesperson's case, the concept of conditional probability is utilized when examining the chance of a sale occurring on the callback, given that no sale was made initially.- This is represented as \( P(S2|NS1) = 0.55 \)- It means the chance of making a sale on the second contact is 55%, provided that the first attempt did not succeed.In fact, this was central to our problem because it wasn't merely about gross probabilities but analyzing how specific scenarios affect the likelihood of different outcomes, allowing for more strategic business decisions.Comprehending conditional probabilities allows one to adjust expectations and strategies as situations evolve during events. It essentially lets us tailor our expectations based on dynamic conditions rather than static measures alone.