Problem 161 - Applied Statistics and Probability for Engineers

Question

A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit the site. Let \(A\) denote the event that the design color is red, and let \(B\) denote the event that the font size is not the smallest one. Are \(A\) and \(B\) independent events? Explain why or why not.

Step-by-Step Solution

Answer

Events \(A\) and \(B\) are independent because their combined probability equals the product of their individual probabilities.

Step 1: Determine Total Possible Designs

To find all possible combinations of web ad designs, multiply the number of choices for each attribute: colors, font types, font sizes, images, and text phrases. This gives us the total number of possible designs. Thus, we have: \[ 4 \text{ colors} \times 3 \text{ fonts} \times 5 \text{ sizes} \times 3 \text{ images} \times 5 \text{ phrases} = 900 \text{ possible designs.} \]

Step 2: Calculate Probability of Event A

Event \(A\) is the event that the design color is red. Since there is only one red color out of four, the probability of \(A\) is: \[ P(A) = \frac{1}{4}. \]

Step 3: Calculate Probability of Event B

Event \(B\) is the event that the font size is not the smallest one. There are five font sizes, so selecting any size other than the smallest gives us four options. The probability of \(B\) is: \[ P(B) = \frac{4}{5}. \]

Step 4: Calculate Probability of A and B Occurring Together

To find the probability of both events \(A\) and \(B\) occurring together (i.e., the ad is red and the font size is not the smallest), multiply the probability of choosing red and one of the non-smallest font sizes. Since these choices are independent in terms of how they are combined (color does not affect font size), we use the product of the probabilities: \[ P(A \cap B) = \left(\frac{1}{4}\right)\times\left(\frac{4}{5}\right) = \frac{1}{5}. \]

Step 5: Check for Independence of Events A and B

Two events are independent if \( P(A \cap B) = P(A) \cdot P(B) \). From earlier steps, we have \( P(A \cap B) = \frac{1}{5} = \frac{1}{4} \times \frac{4}{5} \). This matches the expected result for independent events based on multiplication probability. Thus, events \( A \) and \( B \) are independent.

Key Concepts

Event Independence

Events are considered independent if the occurrence of one event does not affect the probability of the other. In other words, two events, say \( A \) and \( B \), are independent if the probability of both of them occurring simultaneously, \( P(A \cap B) \), is equal to the product of their individual probabilities, \( P(A) \) and \( P(B) \). Therefore, the key equation for independence is:

• Your independent event equation is: \( P(A \cap B) = P(A) \times P(B) \)

In our example, consider event \( A \) as choosing a red color and event \( B \) as choosing a font size that is not the smallest. Since \( P(A \cap B) = \frac{1}{5} \) exactly matches \( P(A) \times P(B) \), we can confidently say that these events are independent. This means the choice of color doesn't influence the choice of font size in any way.

Combinatorial Probability

Combinatorial probability focuses on finding the chance of specific arrangements or combinations of a set occurring. By understanding how many ways a task can be completed using different available options, we can calculate probabilities effectively.
In the case of web ad designs, we consider a variety of choices for each feature:

• 4 possible colors
• 3 kinds of fonts
• 5 sizes of fonts
• 3 image options
• 5 text phrases

To find the total number of unique designs, we multiply the number of available choices for each element. This results in \( 4 \times 3 \times 5 \times 3 \times 5 = 900 \) different combinations, showcasing the essence of combinatorial probability. Each design represents one combination, and the chance of selecting a specific one is a fraction of the total designs.

Probability Calculations

Probability calculations allow us to determine the likelihood of a particular outcome from a number of possibilities. Calculating the probability for specific events involves understanding the event-specific conditions and constraints.
For event \( A \): the design color being red, we divide the favorable outcomes by all outcomes. Since 1 out of the 4 colors is red, \( P(A) = \frac{1}{4} \).
Meanwhile, event \( B \): when the font size is not the smallest, involves choosing from 4 non-smallest sizes out of 5. This yields \( P(B) = \frac{4}{5} \).
When needing to find the probability that both events \( A \) and \( B \) occur together, calculate \( P(A \cap B) \). Assuming independence, this is done by multiplying their individual probabilities: \( \frac{1}{4} \times \frac{4}{5} = \frac{1}{5} \). Such calculations offer insights into the likelihood of multiple conditions being met simultaneously.