Question

Suppose \(P(A)=.1\) and \(P(B)=.5\). a. If \(P(A \mid B)=.1,\) what is \(P(A \cap B) ?\) b. If \(P(A \mid B)=.1,\) are \(A\) and \(B\) independent? c. If \(P(A \cap B)=0,\) are \(A\) and \(B\) independent? d. If \(P(A \cup B)=.65,\) are \(A\) and \(B\) mutually exclusive?

Short answer

Question: Find the probability of the intersection of events A and B, and determine if A and B are independent, mutually exclusive, or if they would be independent if their intersection probability was 0. Answer: The probability of the intersection of events A and B is 0.05. A and B are independent. A and B are not mutually exclusive. If the intersection probability was 0, A and B would not be independent.

Step-by-step solution

a. Find \(P(A \cap B)\)

We are given \(P(A)=0.1\), \(P(B)=0.5\), and \(P(A \mid B)=0.1\). We can find \(P(A \cap B)\) using the formula for conditional probability: \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\). Rearranging to solve for \(P(A \cap B)\), we get \(P(A \cap B) = P(A \mid B) \cdot P(B) = 0.1 \cdot 0.5 = 0.05\).

b. Check if \(A\) and \(B\) are independent

Two events are considered independent if \(P(A \cap B) = P(A) \cdot P(B)\). We found in part (a) that \(P(A \cap B) = 0.05\). Since \(P(A) \cdot P(B) = 0.1 \cdot 0.5 = 0.05\), \(A\) and \(B\) are independent.

c. If \(P(A \cap B)=0\), check if \(A\) and \(B\) are independent

Using the same definition of independence as part (b), we have to check if \(P(A \cap B) = P(A) \cdot P(B)\). Since \(P(A) \cdot P(B) = 0.1 \cdot 0.5 = 0.05 \neq 0\), \(A\) and \(B\) would not be independent if \(P(A \cap B) = 0\).

d. Check if \(A\) and \(B\) are mutually exclusive

Two events are considered mutually exclusive if there's no outcome that is in both events, which means \(P(A \cap B) = 0\). We are given that \(P(A \cup B) = 0.65\). Using the formula \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\), we get \(0.65 = 0.1 + 0.5 - P(A \cap B)\). Solving for \(P(A \cap B)\), we get \(P(A \cap B) = 0.05\). Since \(P(A \cap B) \neq 0\), the events \(A\) and \(B\) are not mutually exclusive.

Key concepts

Independence of Events

When we talk about events being independent, we refer to two events that do not affect each other's occurrence. To put it simply, the happening of one event does not change the probability of the other event happening. In mathematical terms, this means that for two events, \( A \) and \( B \), the condition of independence holds if \( P(A \cap B) = P(A) \cdot P(B) \). This is a crucial concept in probability because it allows us to simplify calculations under specific conditions where events are independent. In our original exercise, we calculated \( P(A \cap B) = 0.05 \) and verified that \( P(A) \cdot P(B) = 0.1 \times 0.5 = 0.05 \). Since these values are equal, events \( A \) and \( B \) are independent in the given scenario. Such understanding helps in solving more complex problems where you have numerous independent events, making the overall probability calculations easier.

• Independence means no influence between events.
• Formula: \( P(A \cap B) = P(A) \cdot P(B) \).
• Example: Our step-by-step solution confirmed \( A \) and \( B \) are independent.

Mutual Exclusivity

Mutual exclusivity is a basic yet critical aspect of probability, defining two events that cannot happen at the same time. In other words, if one event happens, the other cannot. This is distinctly different from independence. For mutually exclusive events \( A \) and \( B \), the rule is \( P(A \cap B) = 0 \). In the original problem, we addressed mutual exclusivity by checking \( P(A \cup B) \) and solving for \( P(A \cap B) \). Given \( P(A \cup B) = 0.65 \), and using the union formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), we found \( P(A \cap B) = 0.05 \). Since \( P(A \cap B) eq 0 \), \( A \) and \( B \) are not mutually exclusive, meaning both events have some overlapping probable outcomes.

• Mutual exclusivity means no overlap between events.
• Formula: \( P(A \cap B) = 0 \).
• Example: Our calculation showed \( A \) and \( B \) are not mutually exclusive.

Probability Calculations

Probability calculations often look complex, but breaking them down makes the task easier. At the heart of it are simple rules and formulas. These calculations help in quantifying the likelihood of different events under various conditions, such as independence and mutual exclusivity.To work through probabilities, understanding key formulas is critical. The conditional probability formula is \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \). For the union of events, it's \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). These allow us to solve problems like finding \( P(A \cap B) \) using known values, as demonstrated in the original exercise.

• Use formulas to simplify calculations.
• Conditional probability helps find intersection probabilities.
• Union of events formula considers overlap corrections.

By mastering these calculations, you can solve a variety of probability problems with confidence and clarity.