Problem 9 - Basic Statistics for Business and Economics with Student

Question

Bank of America customers select their own three-digit personal identification number (PIN) for use at ATMs. a. Think of this as an experiment and list four possible outcomes. b. What is the probability Mr. Jones and Mrs. Smith select the same PIN? c. Which concept of probability did you use to answer (b)?

Step-by-Step Solution

Answer

1/1000 probability they select the same PIN; using classical probability.

Step 1: Understanding the Problem

In this exercise, we are dealing with a three-digit PIN. Each digit can range from 0 to 9. Thus, we are discussing the different possible outcomes and calculate the probability that two different customers choose the same three-digit PIN.

Step 1: List Four Possible Outcomes

Each digit in a PIN is independent and can range from 0 to 9, making 10 options per digit. This results in a total of 1000 possible combinations (ranging from 000 to 999). Four possible outcomes for the PIN could be 123, 456, 789, and 000.

Step 2: Determine the Total Number of Outcomes

Since each digit can be any number from 0 to 9, the number of possible outcomes for the full three-digit PIN is given by multiplying the options for each digit: \(10 \times 10 \times 10 = 1000\).

Step 3: Calculate the Probability of Mr. Jones and Mrs. Smith Choosing the Same PIN

To find the probability that both select the same PIN, we consider the first person selects one PIN (any of the 1000 possibilities), and then the second person must choose the same PIN, which is exactly one specific option out of those 1000 possibilities. Thus, the probability is:\[ \frac{1}{1000} \]

Step 4: Identify the Probability Concept Used

The probability calculated in step 3 employs the classical or theoretical probability concept. This is because each possible outcome (PIN) is equally likely to be chosen, and the specific favorable outcome (both choosing the same PIN) is identified among all possible outcomes.

Key Concepts

Personal Identification Number (PIN)

A Personal Identification Number, commonly known as a PIN, is a numeric code used in the banking sector to authenticate a user's identity. Usually, the PIN is a three or four-digit number that is easy for the user to remember but hard for others to guess. In the context of our exercise, we are focusing on a three-digit PIN.
A three-digit PIN involves selecting numbers from 0 to 9 for each of the three positions. Hence, each digit allows 10 different possibilities (0 through 9).

• 1000 Possible Combinations: Since each digit of the PIN can be any number from 0 to 9, the total number of combinations for a three-digit PIN is calculated by multiplying together the number of possibilities per digit: \(10 \times 10 \times 10 = 1000\).
• Security: The large number of possible combinations makes PINs a secure way to verify identity, assuming no one selects a predictable sequence like "123" or "000".

Understanding how a PIN is constructed and selected is crucial, especially when assessing probability related to such numbers.

Classical Probability

Classical probability, also known as theoretical probability, is an approach used to calculate probabilities based on equally likely outcomes. When talking about classical probability, it implies that every outcome has the same chance of occurring.
In our exercise, the situation deals with calculating the probability that two people select the same three-digit PIN. Classical probability is applied here because:

• Equal Likelihood: Each three-digit PIN, ranging from 000 to 999, has an equal likelihood of being chosen.
• Simple Formula: Probability is calculated using the formula, \( P(A) = \frac{N_f}{N_s} \), where \( P(A) \) is the probability of event A occurring, \( N_f \) is the number of favorable outcomes, and \( N_s \) is the number of possible outcomes.

If each customer's choice of PIN is random and equally likely, the classical probability method states that the chance of Mr. Jones and Mrs. Smith picking the same PIN is \( \frac{1}{1000} \).

Independent Events

In probability, two events are considered independent if the occurrence of one does not affect the occurrence of the other. This concept is fundamental in calculating the probability where the decisions of individuals are concerned.
Relating to our PIN selection exercise, the choice of a PIN by Mr. Jones is independent of the choice by Mrs. Smith.

• Non-Interference: Mr. Jones selecting a particular PIN does not influence which PIN Mrs. Smith selects.
• Calculation: The probability that they pick the same PIN does not change regardless of the number first selected by Mr. Jones. Hence, the chance remains \( \frac{1}{1000} \), as each person has 1000 choices independently.

Understanding independent events helps clarify scenarios where separate choices lead to combined probabilities without mutual impact.