Problem 17 - Calculus for Biology and Medicine

Question

Assume that \(P(A \cap B)=0.1, P(A)=0.4\), and \(P\left(A^{c} \cap B^{c}\right)=\) 0.2. Find \(P(B)\).

Step-by-Step Solution

Answer

The probability of event B occurring is 0.5.

Step 1: Understanding the Problem

We need to find the probability of event B occurring, given probabilities involving events A, B, and their complements. These are: \(P(A \cap B) = 0.1\), \(P(A) = 0.4\), and \(P(A^c \cap B^c) = 0.2\).

Step 2: Using the Complement Rule

Recall that \(P(A) + P(A^c) = 1\). We know \(P(A) = 0.4\), so \(P(A^c) = 1 - 0.4 = 0.6\).

Step 3: Applying the Formula for Complements

Using the complement formula, note that \(P(A^c \cup B^c) = 1 - P(A \cap B)\). We know \(P(A \cap B) = 0.1\), hence \(P(A^c \cup B^c) = 1 - 0.1 = 0.9\).

Step 4: Relationship Between Complements and Intersections

Utilize the formula \(P(A^c \cup B^c) = P(A^c) + P(B^c) - P(A^c \cap B^c)\). We have \(P(A^c \cap B^c) = 0.2\), so \(P(A^c) + P(B^c) - 0.2 = 0.9\).

Step 5: Solving for \(P(B^c)\)

Substitute \(P(A^c) = 0.6\) and simplify: \(0.6 + P(B^c) - 0.2 = 0.9\). Simplifying gives:\(P(B^c) = 0.5\).

Step 6: Finding \(P(B)\)

Since \(P(B) + P(B^c) = 1\), we now find \(P(B) = 1 - 0.5 = 0.5\).

Key Concepts

Complement Rule

In probability theory, the Complement Rule is a fundamental concept that helps us cover all possible scenarios regarding a certain event.
Let's say we have an event A. The complement of A, denoted as \(A^c\), is the event that A does not occur.
The rule is expressed in the simple formula:

• \(P(A) + P(A^c) = 1\)

This formula signifies that the likelihood of an event happening and not happening must add up to 1 because they encompass the entire set of possible outcomes.
Consider if \(P(A)\) is known, like in our exercise where \(P(A) = 0.4\).
You can easily calculate the complement, \(P(A^c)\), using the formula:

• \(P(A^c) = 1 - P(A) = 1 - 0.4 = 0.6\)

This simple relationship is pivotal in solving problems where complements are involved.

Intersection of Events

The intersection of two events A and B, denoted by \(A \cap B\), is the set of outcomes that are common to both events. It's like finding the overlap between two circles in a Venn diagram.
This concept tells us the probability of both events happening simultaneously.
In our exercise, \(P(A \cap B) = 0.1\). This is the probability that both events A and B occur at the same time.
The intersection is a crucial concept when dealing with combined probabilities, because it helps us understand how two events interact. In real-world scenarios, this might include the overlap between two sets of circumstances, like the probability of it raining and being a weekday.

Probability of Complement

Calculating the probability of a complement is a straightforward process once you understand the Complement Rule.
In problems involving intersections, knowing the probability of one complement can help solve for others, just as in our exercise.

• You start by knowing \(P(A^c \cap B^c)\), which is the probability that neither event A nor event B occurs.
• The exercise gives this as \(P(A^c \cap B^c) = 0.2\).
  This probability can help you find other probabilities using other relationships like unions and intersections.

For example, solving for a single complement of another event, like \(P(B^c)\), often involves manipulating known formulas, such as:

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

Using the given probablities, we find that \(P(B^c)\) can be determined, leading to the calculation of \(P(B)\).
This demonstrates the power of understanding compound probability concepts, aiding in the step-by-step approach to complete more complex probability challenges.