Question

A computer salesperson makes either one or two sales contacts each day between 1 and 2 PM. If only one contact is made, the probability is 0.2 that a sale will result and 0.8 that no sale will result. If two contacts are made, the two customers make their decisions independently of each other, each purchasing with probability 0.2 and not purchasing with probability 0.8 . What is the probability that the salesperson has made two sales this hour?

Short answer

The probability of making two sales this hour is 0.04 when two contacts are made.

Step-by-step solution

Analyze Possible Outcomes

We need to determine the probability of the salesperson making two sales in one hour. There are two possible scenarios: \(1\) one contact is made, or \(2\) two contacts are made during the hour.

Calculate Probability for One Contact

If only one contact is made, the probability of making a sale (and thus two sales are impossible) is given as 0.2, and the probability of no sale is 0.8. Therefore, the probability of making two sales from just one contact is 0.

Calculate Probability for Two Contacts

If two contacts are made, each contact results in a sale with a probability of 0.2, independently of each other. The probability of making a sale for each contact is thus \(0.2 \times 0.2 = 0.04\). This represents both sales occurring.

Total Probability for Two Sales

Only the scenario of making two contacts allows for two sales. Therefore, the probability of making two sales this hour can only come from having made two contacts. The probability of two contacts cannot be directly computed from given data but assumed possible (disregarding external probabilities, as they are not provided), leading to the calculated two-contact scenario only.

Key concepts

Discrete Mathematics

Discrete mathematics is a branch of mathematics dealing with countable, distinct elements. In many cases, it focuses on the study of mathematical structures such as sets, graphs, and finite state machines, which are fundamental in computer science, decision-making, and optimization.
Within this exercise, discrete mathematics plays a role in understanding how the salesperson can either make one or two contacts. These are distinctly separate possibilities, which can be thought of as different events. Each event has a specific probability associated with it, representative of discrete outcomes the salesperson might encounter between 1 and 2 PM.
Understanding how probabilities interplay in discrete events is essential. The principle of discrete probabilities indicates that the sum of probabilities for all possible outcomes must equal 1. Therefore, examining all possible scenarios and assigning logical probabilities is a core element of solving such problems. This method allows us to form a basis for calculating precise outcomes like making two sales from two contacts.

Independent Events

Independent events are a fundamental concept in probability theory. Two events are considered independent if the occurrence of one event does not affect the occurrence of the other. In simpler terms, the likelihood of one event occurring remains unchanged by the other event.
In the scenario presented in the exercise, each customer's decision to make a purchase is independent of the other. This means that one customer deciding to buy does not influence the other's decision. Because these events are independent, the probability of both customers making a purchase (i.e., two sales) is the product of the probabilities of each making a purchase.
In this case, with each individual sale having a likelihood of 0.2, the probability of both making a sale is \( P(A \text{ and } B) = P(A) \times P(B) = 0.2 \times 0.2 = 0.04 \).This multiplication rule for independent events makes it straightforward to determine combined probabilities in situations where events do not impact each other, such as two customers deciding separately on a purchase.

Probability Calculation

Probability calculation involves determining how likely an event is to occur out of all possible outcomes. This is expressed as a fraction, percent, or decimal between 0 and 1. In the context of this exercise, it requires evaluating the likelihood of making two sales within a specific time frame.
Probability calculations often begin with identifying possible outcomes, in this case, the number of contacts made and the resulting sales. After determining all possible scenarios, calculations should consider any dependencies or independencies among events.
The calculation for two sales from two contacts in our exercise is based on both contacts independently leading to a purchase. We've already calculated the probability of each sale (0.2 each). By taking the product of these independent probabilities, \(0.2 \times 0.2 = 0.04\), we determine the probability of both events occurring simultaneously.Since only the two contacts scenario allows for two sales, the solution highlights the necessity of calculating probabilities for each distinct scenario and summing probabilities if applicable. Here, we only need to consider the two-contact scenario where two sales are possible. Hence, calculations for probability should always anchor to identified potential outcomes bounded by any given constraints, such as time or number of contacts in this problem.