Problem 210 - Applied Statistics and Probability for Engineers

Question

The British government has stepped up its information campaign regarding foot- and-mouth disease by mailing brochures to farmers around the country. It is estimated that \(99 \%\) of Scottish farmers who receive the brochure possess enough information to deal with an outbreak of the disease, but only \(90 \%\) of those without the brochure can deal with an outbreak. After the first three months of mailing, \(95 \%\) of the farmers in Scotland had received the informative brochure. Compute the probability that a randomly selected farmer will have enough information to deal effectively with an outbreak of the disease.

Step-by-Step Solution

Answer

98.55%

Step 1: Understand Given Probabilities

We are given that 95% of Scottish farmers received the brochure and 99% of those farmers can deal with an outbreak. Meanwhile, 5% did not receive the brochure, and 90% of those farmers can deal with an outbreak.

Step 2: Define Events and Probabilities

Let Event A be that a farmer receives the brochure. Prob(A) = 0.95. Let Event B be that a farmer can handle an outbreak. We need to find Prob(B). We know: Prob(B|A) = 0.99 and Prob(B|A') = 0.90, where A' is the complement of A.

Step 3: Use Total Probability Formula

The total probability formula states: Prob(B) = Prob(B|A) * Prob(A) + Prob(B|A') * Prob(A'). Substitute the given values: - Prob(B|A) = 0.99 - Prob(A) = 0.95 - Prob(B|A') = 0.90 - Prob(A') = 1 - Prob(A) = 0.05. So, Prob(B) = (0.99 * 0.95) + (0.90 * 0.05).

Step 4: Calculate Each Component

Calculate the first term: 0.99 * 0.95 = 0.9405. Then calculate the second term: 0.90 * 0.05 = 0.045.

Step 5: Sum the Components

Add the two components together to obtain the total probability: 0.9405 + 0.045 = 0.9855.

Step 6: Result Interpretation

The probability that a randomly selected farmer will have enough information to deal with an outbreak of foot-and-mouth disease is 98.55%.

Key Concepts

Probability Theory

Probability theory is a branch of mathematics concerned with the analysis of random phenomena. It provides the foundational framework for handling uncertainties and making predictions about outcomes. In essence, probability theory quantifies how likely events are to occur.

The probability of an event is a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. For example, in this exercise, we determine the probability a farmer is equipped with enough information to handle an outbreak effectively. By applying probability theory, we gather insights not just into chance occurrences, but how to prepare for them with the data available.

Conditional Probability

Conditional probability refers to the likelihood of an event occurring, given that another event has already occurred. It's a crucial concept in situations where events are interdependent. In this exercise, we use conditional probability when considering whether a farmer can deal with an outbreak, contingent upon whether they received the brochure.

By defining Event A as receiving the brochure and Event B as being able to handle an outbreak, conditional probabilities are expressed as \( \text{Prob}(B|A) \) and \( \text{Prob}(B|A') \). Here, \( \text{Prob}(B|A) = 0.99 \) entails that 99% of brochure-receiving farmers are prepared, while \( \text{Prob}(B|A') = 0.90 \) relates to the 90% readiness of those who did not receive it.

Conditional probabilities allow us to compute realistic expectations by considering one condition's effect on another.

Event Outcomes

In the realm of probability, event outcomes signify the possible results of a given scenario. Understanding event outcomes is crucial as it helps in framing the proper probabilities for complex analyses. In this context, an event could be either a farmer receiving or not receiving the brochure, and their subsequent capability to handle the outbreak.

Defining events precisely allows for an accurate configuration of dependencies and outcomes. For this exercise, computing the probability involves recognizing these outcomes:

• Receiving the brochure and being prepared for the outbreak
• Not receiving the brochure but still being prepared

Analyzing these possible outcomes lays the groundwork for calculating more intricate probabilities using the theoretical underpinnings from probability concepts.

Statistical Analysis

Statistical analysis is the practice of collecting and examining data to discern patterns and draw meaningful conclusions. It bridges the gap between raw data and insightful conclusions. In our example, statistical analysis combines various probability components to find the overall likelihood of farmers being prepared for an outbreak.

Key to statistical analysis in this context is the use of the Total Probability Theorem. This allows us to integrate probabilities from different grouped outcomes to reach a comprehensive understanding. We calculate \[ \text{Prob}(B) = \text{Prob}(B|A) \cdot \text{Prob}(A) + \text{Prob}(B|A') \cdot \text{Prob}(A') \]for our overall preparation probability.

Through this analysis, we conclude with the probability of 98.55%, showcasing the power of combined probabilistic modeling and statistical analysis.