Problem 25 - Basic Statistics for Business and Economics with Student

Question

A local bank reports that 80 percent of its customers maintain a checking account, 60 percent have a savings account, and 50 percent have both. If a customer is chosen at random, what is the probability the customer has either a checking or a savings account? What is the probability the customer does not have either a checking or a savings account?

Step-by-Step Solution

Answer

The probability of having either account is 0.90; not having either is 0.10.

Step 1: Identify Given Probabilities

We are given the following probabilities: \( P(C) = 0.80 \) for a checking account, \( P(S) = 0.60 \) for a savings account, and \( P(C \cap S) = 0.50 \) for having both.

Step 2: Calculate Probability of Either Account

Using the formula for the probability of either event occurring, \( P(C \cup S) = P(C) + P(S) - P(C \cap S) \), substitute the given values: \( P(C \cup S) = 0.80 + 0.60 - 0.50 = 0.90 \). Hence, the probability of a customer having either a checking or a savings account is 0.90.

Step 3: Calculate Probability of Neither Account

Since the probability of having either a checking or a savings account is \( P(C \cup S) = 0.90 \), the probability of not having either is the complement: \( 1 - P(C \cup S) = 1 - 0.90 = 0.10 \). Therefore, the probability that a customer does not have either type of account is 0.10.

Key Concepts

Set Theory

Set theory is a fundamental concept in probability theory that helps us understand how groups of items interact with each other. It is like organizing different items into containers called "sets," where each container holds a group of items with common attributes. These sets allow us to define different operations like union, intersection, and complement, which are essential for calculating probabilities.

In the context of our bank example, we have sets defined for customers with a checking account and customers with a savings account. Set theory helps us visualize these customer groups and perform calculations based on their overlap and differences. Understanding these interactions offers a clear picture of the relationship between sets and the chances of various outcomes occurring. By defining sets in terms of real-world data, such as account ownership, set theory becomes a practical tool for representing probabilistic events.

Complementary Events

Complementary events are two outcomes that account for all possible outcomes of an experiment or observation. They are two sides of the same coin: if one event occurs, the other does not. The probabilities of complementary events add up to 1, expressing the certainty that one outcome from the pair will happen.

In our exercise, the events are having a checking or savings account, and not having any account. These are complements because the occurrence of one directly implies the non-occurrence of the other. In probability terms, if we know that the probability of having either account is 0.90, the complementary event—having neither account—has a probability of 1 minus 0.90, which is 0.10. This concept is vital for calculating unseen probabilities from known data, making it easier to cover all possibilities in probability analyses.

Intersection and Union in Probability

The intersection and union of sets are two key operations in probability theory. They help determine how likely two or more events are to occur together or separately.

The **intersection** of two sets, represented as \( P(C \cap S) \), refers to the probability that both events occur simultaneously. For the bank example, this is the probability (0.50) of customers having both a checking and a savings account. Intersection is crucial for understanding overlapping events and shared outcomes within different sets.

The **union** of sets, denoted as \( P(C \cup S) \), signifies the probability of at least one of the events occurring. Our solution seeks this probability, calculated as \( P(C) + P(S) - P(C \cap S) = 0.90 \), meaning the likelihood of a customer having either a checking or savings account. These operations reflect the additive and multiplicative nature of probability, similar to combining or comparing datasets to assess collective probabilities. By mastering these concepts, one can evaluate complex scenarios where multiple variables interact.