Problem 16 - Calculus for Biology and Medicine

Question

Assume that \(P\left(A \cap B^{c}\right)=0.1, P\left(B \cap A^{c}\right)=0.5\), and \(P\left((A \cup B)^{c}\right)=0.2\). Find \(P(A \cap B)\).

Step-by-Step Solution

Answer

The probability \( P(A \cap B) \) is 0.2.

Step 1: Understand Set Theory Notation

The problem involves different set operations using complements and intersections of two events, A and B. Here's the breakdown:- \( P(A \cap B^c) \) is the probability of A happening and B not happening.- \( P(B \cap A^c) \) is the probability of B happening and A not happening.- \( P((A \cup B)^c) \) is the probability of neither A nor B occurring. We aim to find \( P(A \cap B) \), the probability of both A and B occurring.

Step 2: Apply De Morgan’s Law for Complements

We have \( P((A \cup B)^c) = 0.2 \). By De Morgan’s Law, \( P((A \cup B)^c) = 1 - P(A \cup B) \). Therefore, \( P(A \cup B) = 1 - 0.2 = 0.8 \).

Step 3: Use the Inclusion-Exclusion Principle

The probability of the union of A and B is given by:\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]We aim to find \( P(A \cap B) \).

Step 4: Express Union with Known Probabilities

We know from the given problem:\[ P(A \cap B^c) = 0.1 \]\[ P(B \cap A^c) = 0.5 \]By the inclusion-exclusion principle: \[ P(A) = P(A \cap B) + P(A \cap B^c) \]\[ P(B) = P(B \cap A) + P(B \cap A^c) \]

Step 5: Find P(A) and P(B)

The expressions become:\[ P(A) = P(A \cap B) + 0.1 \]\[ P(B) = P(A \cap B) + 0.5 \]

Step 6: Substitute Union Expression Values

Substitute into the union expression:\[ 0.8 = (P(A \cap B) + 0.1) + (P(A \cap B) + 0.5) - P(A \cap B) \]

Step 7: Simplify and Solve for P(A ∩ B)

Simplify the equation:\[ 0.8 = 0.1 + 0.5 + P(A \cap B) \]\[ 0.8 = 0.6 + P(A \cap B) \]\[ P(A \cap B) = 0.8 - 0.6 \]Thus, \( P(A \cap B) = 0.2 \).

Key Concepts

Set Operations

In probability theory, understanding set operations is crucial for accurately determining probabilities in scenarios involving multiple events. Set operations help us visualize and calculate probabilities associated with various intersections, unions, and complements of events.
Let's break down some basic set operations:

• Intersection (\(A \cap B\)): This operation represents the probability that both events A and B occur. It's like finding the common elements in two sets.
• Union (\(A \cup B\)): This involves the probability that either event A, event B, or both, occur. It's the combined area of two sets.
• Complement (\(A^c\)): This operation refers to the probability that event A does not occur. It's like considering everything outside the set A.

Additionally, Venn diagrams are useful tools for visually representing set operations in probability problems. They allow us to see how different events overlap and differ, helping us make sense of equations involving these operations.
Mastering set operations will make it easier to tackle more complex probability problems.

Inclusion-Exclusion Principle

When dealing with probabilities involving multiple events, the inclusion-exclusion principle becomes immensely helpful. It allows us to calculate the probability of the union of two or more events by accounting for overlaps.
Here's a simple view of how it works for two events, A and B: - The probability of event A or event B happening is given by \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]- The term \(P(A \cap B)\) is subtracted because it is counted twice when adding \(P(A)\) and \(P(B)\).

• This principle ensures that any overlapping probabilities between events aren't counted more than once.
• It becomes particularly important in complex problems where more than two events are involved, preventing over-counting.

By understanding and appropriately applying the inclusion-exclusion principle, we can break down and solve probability problems involving unions effectively, as was demonstrated in our original problem.

De Morgan's Law

De Morgan's Laws are two fundamental rules in probability and set theory that relate to complements and unions/intersections of sets. They are crucial in understanding how to transform complex probabilities into more manageable ones.
The two laws are:

• First Law: The complement of the union of two sets is the intersection of their complements: \[ (A \cup B)^c = A^c \cap B^c \]
• Second Law: The complement of the intersection of two sets is the union of their complements: \[ (A \cap B)^c = A^c \cup B^c \]

In our exercise, the first law was used to find the probability of the union of A and B by transforming and working through its complement probability.
This helped us transition from complex event probabilities into simpler terms, specifically allowing us to manipulate these expressions with greater flexibility.
Ultimately, using De Morgan's Laws in probability calculations allows for a deeper understanding of how events interact, especially when dealing with complements.