Problem 12 - Calculus for Biology and Medicine

Question

Assume that \(\Omega=\\{1,2,3,4,5\\}\) \(P(1)=0.1, P(2)=0.2\), and \(P(3)=P(4)=0.05 .\) Furthermore, assume that \(A=\\{1,3,5\\}\) and \(B=\\{2,3,4\\}\) Find \(P(A \cup B)\)

Step-by-Step Solution

Answer

The probability \(P(A \cup B)\) is 0.4.

Step 1: Understanding the Event Union

To find the probability of the union of two events, say \(A \cup B\), we need to ensure that we include the probability of elements from both events without double-counting any element that might appear in both events. Mathematically, this is expressed as:\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

Step 2: Calculating Probability of Event A

Event \(A\) consists of elements \(\{1, 3, 5\}\). According to given probabilities, the probability of \(A\) is:\[P(A) = P(1) + P(3) + P(5) = 0.1 + 0.05 + 0 = 0.15\]Note: Since the probability of element 5 is not given, we assume it to be 0.

Step 3: Calculating Probability of Event B

Event \(B\) consists of elements \(\{2, 3, 4\}\). The probability of \(B\) is:\[P(B) = P(2) + P(3) + P(4) = 0.2 + 0.05 + 0.05 = 0.3\]

Step 4: Finding the Intersection of A and B

The intersection \(A \cap B\) contains only element 3, which is common in both \(A\) and \(B\). Therefore, the probability is:\[P(A \cap B) = P(3) = 0.05\]

Step 5: Applying the Union Formula

Substitute the computed probabilities into the union formula:\[P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.15 + 0.3 - 0.05 = 0.4\]

Step 6: Conclusion

The probability of the union of events \(A\) and \(B\) is thus calculated to be 0.4, meaning there's a 40% chance that an element in either event \(A\) or event \(B\) occurs.

Key Concepts

Set Operations

In probability theory, set operations are essential for understanding how different events relate to each other. Events in probability are similar to sets in mathematics, which are simply collections of outcomes. These operations help us calculate probabilities in various circumstances.

There are three primary set operations:

• Union: Combines elements from multiple events.
• Intersection: Identifies common elements in events.
• Complement: Consists of elements not in the event.

Understanding set operations allows us to effectively gauge the likelihood of various outcomes. Each operation has specific rules and formulas that describe event relationships, making them crucial for probability calculations.

Union of Events

The concept of the union of events is fundamental when working with probabilities. The union of two events, denoted as \(A \cup B\), represents the event where any outcome from either \(A\) or \(B\) (or both) occurs. It's like pooling together the elements from both sets.

For example, if event \(A\) is \(\{1, 3, 5\}\) and event \(B\) is \(\{2, 3, 4\}\), the union \(A \cup B\) would include \(\{1, 2, 3, 4, 5\}\). Notice how each unique element from both \(A\) and \(B\) is included just once. This is crucial as it prevents over-counting, which can skew probability results.

• Formula for Union: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

This formula correctly accounts for elements appearing in both events by subtracting the intersection's probability.

Intersection of Events

The intersection of events \(A\) and \(B\), symbolized as \(A \cap B\), includes only the outcomes common to both \(A\) and \(B\). It's similar to finding the overlap between two sets.

For example, in our given exercise, \(A\) is \(\{1, 3, 5\}\) and \(B\) is \(\{2, 3, 4\}\). Their intersection would be \(\{3\}\) because 3 is the only number present in both sets. This intersection is crucial for correctly calculating the probability of the union, helping us to avoid double counting common elements.

• Importance: Subtracts overlap from probability calculations.

Finding intersections ensures all elements in both events are uniquely and accurately represented in probability calculations.

Probability Calculations

Calculating probabilities involves using set operations to determine the likelihood of an event. In the original problem, different formulas and steps were used to find \(P(A \cup B)\). Here's a simple breakdown of how probability calculations are performed:

• Identify Elements: List all elements in \(A\) and \(B\).
• Find Probabilities: Calculate probabilities using given data, accounting for all elements.
• Apply Union Formula: Use \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\) to find the likelihood of either event occurring.

In this exercise, with probabilities calculated as \(P(A)=0.15\), \(P(B)=0.3\), and \(P(A \cap B)=0.05\), the probability of union \(P(A \cup B)\) simplifies to 0.4. Probability calculations are the backbone of probability theory, helping us understand the likelihood of various events within a given sample space.