Problem 61 - Applied Statistics and Probability for Engineers

Question

If the last digit of a weight measurement is equally likely to be any of the digits 0 through 9 (a) What is the probability that the last digit is \(0 ?\) (b) What is the probability that the last digit is greater than or equal to \(5 ?\)

Step-by-Step Solution

Answer

(a) 0.1 (b) 0.5

Step 1: Understand the Problem

We are given that the last digit of a weight measurement, ranging from 0 to 9, is equally likely to be any of these digits. We need to calculate the probabilities for specific outcomes based on this information.

Step 2: Calculate Probability for Digit Being 0

Since each digit from 0 to 9 is equally likely, the probability of the last digit being 0 is simply the reciprocal of the total number of possible digits. There are 10 digits in total.Thus, the probability that the last digit is 0 is given by \[ P(\text{digit is 0}) = \frac{1}{10} = 0.1 \]

Step 3: Identify Digits Greater than or Equal to 5

We need to determine which digits, from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, are greater than or equal to 5. These digits are {5, 6, 7, 8, 9}. This gives us a total of 5 digits.

Step 4: Calculate Probability for Digit Being Greater than or Equal to 5

The probability of selecting a digit greater than or equal to 5 is the number of favorable outcomes divided by the total number of possible outcomes.The probability that the digit is greater than or equal to 5 is \[ P(\text{digit} \geq 5) = \frac{5}{10} = 0.5 \]

Key Concepts

Equally Likely Events

In probability, an event is described as "equally likely" if every possible outcome has the same chance of occurring. This is a foundational concept necessary to understand when comparing probabilities in simple scenarios like rolling dice or drawing cards. In our exercise, each digit from 0 to 9 has an equal chance of being the last digit of a weight measurement.

• Each digit is considered equally probable.
• This results in a uniform probability distribution.

The concept ensures that our calculations start from a level playing field by assuming no inherent bias for any digit. This assumption simplifies the process, making it straightforward to compute probabilities for specific outcomes.

Probability Calculation

Finding the probability of an event involves determining how many ways that event can occur and dividing it by the total number of possible outcomes. This is done under the assumption of equally likely outcomes. The formula for computing such a probability is:

• \[ P( ext{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

Using this formula:

• For a digit to be "0", our favorable outcome is "0" itself, resulting in a probability of \( \frac{1}{10} = 0.1 \).
• For digits greater than or equal to "5" (i.e., 5, 6, 7, 8, 9), there are 5 favorable outcomes. Hence, the probability is \( \frac{5}{10} = 0.5 \).

Calculating probability lets us quantify the likelihood of future events, assisting in decision-making processes based on statistical information.
Understanding calculations is helpful in various fields beyond math, fostering analytical and critical thinking.

Discrete Outcomes

When discussing probabilities, it’s essential to distinguish between discrete and continuous outcomes. Discrete outcomes are finite or countable. In the context of our exercise, the digit being any number between 0 and 9 is naturally a discrete outcome.

• Each outcome is specific and distinct.
• The number of outcomes does not change dynamically.

This differs significantly from continuous outcomes, where possibilities form an entire range or spectrum, like measuring time or height. With discrete outcomes, we can assign clear, distinct probabilities to each event, which simplifies calculations and predictions. This is particularly useful in scenarios like this exercise, where we are working with finite sets.