Problem 9 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

A certain grapefruit variety is grown in two regions in southern Spain. Both areas get infested from time to time with parasites that damage the crop. Let \(A\) be the event that region \(R_{1}\) is infested with parasites and \(B\) that region \(R_{2}\) is infested. Suppose \(\mathrm{P}(A)=3 / 4, \mathrm{P}(B)=2 / 5\) and \(\mathrm{P}(A \cup B)=4 / 5\). If the food inspection detects the parasite in a ship carrying grapefruits from \(R_{1}\), what is the probability region \(R_{2}\) is infested as well?

Step-by-Step Solution

Answer

The probability that region \( R_2 \) is infested given region \( R_1 \) is infested is \( \frac{7}{15} \).

Step 1: Define the Problem

We need to find the probability that region \( R_2 \) is infested given that region \( R_1 \) is infested, i.e., \( \mathrm{P}(B|A) \).

Step 2: Use the Formula for Conditional Probability

Conditional probability \( \mathrm{P}(B|A) \) is given by the formula: \[ \mathrm{P}(B|A) = \frac{\mathrm{P}(A \cap B)}{\mathrm{P}(A)} \]. We know \( \mathrm{P}(A) = \frac{3}{4} \), but we need to find \( \mathrm{P}(A \cap B) \) first.

Step 3: Use the Formula for Union of Events

The probability of the union of events \( A \) and \( B \) is given by: \[ \mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B) \]. We have that \( \mathrm{P}(A \cup B) = \frac{4}{5} \), \( \mathrm{P}(A) = \frac{3}{4} \), and \( \mathrm{P}(B) = \frac{2}{5} \).

Step 4: Solve for Intersection Probability

Rearranging the formula from Step 3: \[ \mathrm{P}(A \cap B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cup B) \]. Substituting the values: \[ \mathrm{P}(A \cap B) = \frac{3}{4} + \frac{2}{5} - \frac{4}{5} \].

Step 5: Calculate Intersection Probability

Convert \( \frac{3}{4} \) to \( \frac{15}{20} \), and \( \frac{2}{5} \) to \( \frac{8}{20} \), and \( \frac{4}{5} \) to \( \frac{16}{20} \). Thus, \[ \mathrm{P}(A \cap B) = \frac{15}{20} + \frac{8}{20} - \frac{16}{20} = \frac{7}{20} \].

Step 6: Apply Conditional Probability Formula

Now substitute \( \mathrm{P}(A \cap B) = \frac{7}{20} \) and \( \mathrm{P}(A) = \frac{3}{4} = \frac{15}{20} \) into the conditional probability formula: \[ \mathrm{P}(B|A) = \frac{7/20}{15/20} = \frac{7}{15} \].

Key Concepts

Probability of Intersection

When dealing with probabilities, understanding the probability of the intersection of two events is crucial. The intersection of events refers to the event that both events occur. In symbolic form, the intersection is denoted as \(A \cap B\). To find the probability of this intersection, we use the formula:

• \( \mathrm{P}(A \cap B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cup B) \)

This formula finds its use in scenarios where two events, say \(A\) and \(B\), are each associated with probabilities, and their simultaneous occurrence needs to be determined. For instance, in our exercise, both regions \(R_1\) and \(R_2\) getting infested is the intersection event. By plugging in the values:

• \( \mathrm{P}(A) = \frac{15}{20} \) and \( \mathrm{P}(B) = \frac{8}{20} \)
• Subtracting the union \( \mathrm{P}(A \cup B) = \frac{16}{20} \)
• We get \( \mathrm{P}(A \cap B) = \frac{7}{20} \)

This intersection tells us that there's a \( \frac{7}{20} \) probability both regions are infested simultaneously.

Union of Events

The union of events refers to the occurrence of at least one of the events happening. In probability terms, this is denoted by \(A \cup B\). It embodies scenarios such as either event \(A\), event \(B\), or both happening. The probability of the union is given by:

• \( \mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B) \)

This formula captures the likelihood of at least one of the events occurring, while adjusting for any overlap in their occurrence (the intersection).
In our context, \(A\) represents region \(R_1\) being infested, and \(B\) region \(R_2\). From the exercise, the union probability is given as \( \frac{4}{5} \).

• This means there's an 80% chance that at least one of the regions is infested.
• It's important because knowing the extent of overlap helps us pinpoint more precise probabilities like the conditional probability later.

Probability Theory

Probability theory is the mathematical framework that quantifies uncertainty. It equips us with the tools to measure the likelihood of various events. One key concept within this theory is conditional probability, which questions given one event occurring, what is the probability of another event occurring.
Conditional Probability:Conditional probability is calculated using:

• \( \mathrm{P}(B|A) = \frac{\mathrm{P}(A \cap B)}{\mathrm{P}(A)} \)

This formula allows us to refine probabilities based on additional information. In the exercise, this concept answers the question if \(R_1\) is already infested, how likely is \(R_2\) to be infested too?

• Given \( \mathrm{P}(A \cap B) = \frac{7}{20} \) and \( \mathrm{P}(A) = \frac{15}{20} \)
• The conditional probability is \( \mathrm{P}(B|A) = \frac{7}{15} \)

Knowing this tells farmers the likelihood of multiple infestations, helping with more precise risk management.