Problem 34 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The manager of a local bank observes how long it takes a customer to complete his transactions at the automatic bank teller. a. Describe an appropriate sample space for this experiment. b. Describe the event that it takes a customer between 2 and 3 min to complete his transactions at the automatic bank teller.

Step-by-Step Solution

Answer

a. The sample space S for this experiment can be represented as an interval of non-negative numbers, including zero up to positive infinity: \[S = [0, +\infty)\] b. The event E, where a customer takes between 2 and 3 minutes to complete a transaction at the automatic bank teller can be written as: \[E = [2,3]\]

Step 1: Understand the sample space

The sample space is the set of all possible outcomes of an experiment. In this case, the sample space consists of all the possible times it could take a customer to complete a transaction at the automatic bank teller.

Step 2: Represent the sample space

Since the time can be any non-negative number, we can represent the sample space as an interval starting from 0 to positive infinity. Mathematically, the sample space (denoted by 'S') can be written as: \[S = [0, +\infty)\] This means that the sample space includes all non-negative numbers (including zero) up to positive infinity. b. Describe the event that it takes a customer between 2 and 3 min to complete his transactions at the automatic bank teller.

Step 1: Understand the event

An event is a subset of the sample space, representing one or more outcomes that satisfy a certain condition. In this case, the condition is that it takes a customer between 2 and 3 minutes to complete the transaction.

Step 2: Represent the event

The event in question can be represented as a subset of the sample space, in the form of an interval that includes all numbers between 2 and 3 minutes. Mathematically, the event (denoted by 'E') can be written as: \[E = [2,3]\] This means that the event includes all the transaction times between 2 and 3 minutes, inclusive of the endpoints 2 and 3 minutes.

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the analysis of random events. The main objective of this theory is to provide a mathematical foundation for understanding and modeling uncertainty. It is used to predict the likelihood of events occurring, based on a certain set of initial conditions.

Using probability theory, one can quantify the chances of a particular event happening by assigning it a probability value, which is a number between 0 and 1. A probability of 0 indicates impossibility, while a probability of 1 means certainty. For the bank teller example, a probability value would help us understand the chances that a customer takes exactly 2, between 2 and 3, or more than 3 minutes to complete their transaction.

To calculate probabilities, one must have a clear definition of the sample space, which includes all possible outcomes. In the exercise, the sample space is represented as an interval from 0 to positive infinity, denoting all possible transaction times at the automatic bank teller. Once the sample space is established, individual events within that sample space, such as the event where a transaction takes between 2 and 3 minutes, can be analyzed and assigned probabilities.

Applied Mathematics

Applied mathematics involves the application of mathematical methods by different fields such as science, engineering, business, and industry. In our exercise on the banking scenario, applied mathematics would come into play when the manager uses the sample space and probability concepts to improve the efficiency of the banking services.

For instance, understanding the distribution of transaction times can help in staffing decisions, determining the number of tellers needed at different times of the day to minimize customer wait times. Calculating the probabilities of various transaction times could also assist in designing user interfaces that are more efficient, ensuring a higher percentage of transactions fall within a desired timeframe, leading to better customer satisfaction.

Such real-world applications demonstrate the importance of translating abstract mathematical concepts into practical solutions. The study of applied mathematics develops skills in modeling, problem-solving, and reasoning that are essential for tackling complex problems in a wide array of practical domains.

Event in Probability

In probability, an event is a set of outcomes of an experiment to which a probability is assigned. An event can be simple, consisting of a single outcome, or compound, involving multiple outcomes. From our textbook problem, the event is the time interval during which a customer completes transactions at the automatic teller.

Mathematically, this event is represented by the interval \[E = [2,3]\], which captures the essence of what we are investigating: how likely it is for transaction times to fall within this range. When working with events, it's vital to precisely define what we're interested in. This often requires creating a clear, unambiguous representation of our conditions, which in this case is the time interval.

Understanding the nature of events, and their representation in a sample space is a fundamental skill in probability theory. It allows us to analyze and come up with solutions to probabilistic questions that can be both theoretically interesting and practically relevant, such as optimizing service times in a bank.