Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

If \(A\) and \(B\) are independent events, \(P(A)=.4\), and \(P(B)=.6\), find a. \(P(A \cap B)\) b. \(P(A \cup B)\)

Step-by-Step Solution

Answer

a. \(P(A \cap B) = 0.24\) b. \(P(A \cup B) = 0.76\)

Step 1: Find the probability of the intersection

Since A and B are independent events, we can find the probability of their intersection using the formula mentioned above: $$ P(A \cap B) = P(A) \times P(B) $$ Given that \(P(A) = 0.4\) and \(P(B) = 0.6\), we can substitute these values into the formula: $$ P(A \cap B) = 0.4 \times 0.6 $$

Step 2: Calculate the intersection

Now, multiply the probabilities together to get the intersection probability: $$ P(A \cap B) = 0.4 \times 0.6 = 0.24 $$ Therefore, the probability of the intersection of A and B is 0.24.

Step 3: Find the probability of the union

Now, let's find the probability of the union of A and B using the formula mentioned in the analysis section: $$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$ We already know the probabilities of A, B, and the intersection: $$ P(A) = 0.4, P(B) = 0.6, P(A \cap B) = 0.24 $$ Substitute these values into the formula: $$ P(A \cup B) = 0.4 + 0.6 - 0.24 $$

Step 4: Calculate the union

Finally, add and subtract the probabilities to get the union probability: $$ P(A \cup B) = 0.4 + 0.6 - 0.24 = 0.76 $$ Therefore, the probability of the union of A and B is 0.76. So, the answers are: a. \(P(A \cap B) = 0.24\) b. \(P(A \cup B) = 0.76\)

Key Concepts

Independent Events

In probability theory, independent events are two events that do not influence each other. This means that the occurrence of one event does not affect the probability of the other event occurring. A simple real-life example could be tossing a coin and rolling a die. The outcome of the coin toss does not affect the number that appears on the die.

To determine if two events, say event A and event B, are independent, we confirm that the probability of the intersection of these events equals the product of their individual probabilities. Mathematically, this is expressed as:

• \( P(A \cap B) = P(A) \times P(B) \)

This formula is crucial when working with independent events, as it helps us calculate the probability of both events occurring together. If the equality holds, we can confidently say the events are independent.

Intersection of Events

The intersection of events, often symbolized as \( A \cap B \), represents the probability of both event A and event B occurring simultaneously. It's like a shared space where both conditions must be true. For example, if Event A is attending an online class, and Event B is submitting an assignment, \( A \cap B \) would denote scenarios where both attending and submitting occur.

For independent events, the formula to calculate the intersection is straightforward:

• \( P(A \cap B) = P(A) \times P(B) \)

In the exercise provided, with \( P(A) = 0.4 \) and \( P(B) = 0.6 \), the intersection was calculated as \( 0.24 \). This calculation gives a precise measurement of the likelihood of both events happening at the same time.

Union of Events

The union of events, expressed as \( A \cup B \), is the probability that either event A or event B or both occur. It's like the combination of the two event spaces, where either one happening makes the union true. For instance, if an event A is achieving a passing grade, and an event B is winning a scholarship, the union \( A \cup B \) covers scenarios where a student achieves either one or both.

The formula to find the probability of the union of two events is given by:

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

This formula helps avoid double counting the cases where both events happen together, which is included in both \( P(A) \) and \( P(B) \). In our exercise, substituting the known values gives \( 0.76 \) as the probability of the union, showing how often at least one of the events will occur.

Probability Formulas

Probability formulas are essential tools in calculating the chances of various outcomes. They offer a straightforward way to approach complex problems by breaking them down into simpler components.

For independent events, the primary formulas as seen in the exercise include:

• Intersection of Independent Events: \( P(A \cap B) = P(A) \times P(B) \)
• Union of Two Events: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

These formulas are the backbone of many probability problems and enable precise calculations. Understanding them is key to mastering more advanced topics in probability and statistics, as they lay the foundation for logical reasoning and mathematical computation in real-world applications.