Problem 18 - Calculus for Biology and Medicine

Question

Assume that \(P(A)=0.4, P(B)=0.4\), and \(P(A \cup B)=0.7\). Find \(P(A \cap B)\) and \(P\left(A^{c} \cap B^{c}\right)\).

Step-by-Step Solution

Answer

\(P(A \cap B) = 0.1\) and \(P(A^c \cap B^c) = 0.3\).

Step 1: Using the Formula for Union of Two Events

To find the probability of the intersection of two events, we use the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]Substitute the known probabilities into this formula:\[ 0.7 = 0.4 + 0.4 - P(A \cap B) \]

Step 2: Solving for the Intersection Probability

Rearrange the equation from Step 1 to solve for \(P(A \cap B)\):\[ P(A \cap B) = 0.4 + 0.4 - 0.7 \]\[ P(A \cap B) = 0.8 - 0.7 \]\[ P(A \cap B) = 0.1 \]

Step 3: Finding the Complement of the Union

The complement of \(A \cup B\) is \(A^c \cap B^c\). We use the formula:\[ P(A^c \cap B^c) = 1 - P(A \cup B) \]Substitute the given probability:\[ P(A^c \cap B^c) = 1 - 0.7 \]\[ P(A^c \cap B^c) = 0.3 \]

Step 4: Conclusion on Probability Values

We have found the probabilities for both required events:- \(P(A \cap B) = 0.1\)- \(P(A^c \cap B^c) = 0.3\)

Key Concepts

Understanding Intersections of Events

In probability, an intersection refers to the occurrence of two or more events happening simultaneously. Think of it like a Venn diagram where we're interested in the overlapping region representing both events occurring at the same time. The formula for the intersection of two events, denoted as \(P(A \cap B)\), is derived from the formula for the union of events:

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

This equation tells us that the probability of either event A or event B occurring includes both individual probabilities minus the probability of both events occurring simultaneously. It's essential to subtract the intersection once, otherwise, we count the intersection twice when summing \(P(A)\) and \(P(B)\).
For example, if \(P(A)\) and \(P(B)\) are each \(0.4\), the intersection \(P(A \cap B)\) can be calculated by rearranging the formula:

• \(0.7 = 0.4 + 0.4 - P(A \cap B)\)
  \(P(A \cap B) = 0.8 - 0.7 = 0.1\)

This calculation shows that there's a 0.1 probability that both A and B occur together.

Applying the Complement Rule

The complement rule in probability helps us find the probability of the non-occurrence of events. Simply put, it calculates the likelihood that an event does not happen. The complement of an event \(A\) is denoted as \(A^c\) and the rule can be expressed as:

• \(P(A^c) = 1 - P(A)\)

Applying this concept to the union of events, the complement of \(A \cup B\) is represented as \(A^c \cap B^c\), indicating that neither A nor B occur. It is calculated using:

• \(P(A^c \cap B^c) = 1 - P(A \cup B)\)

For example, if the probability of either A or B occurring (their union) is \(0.7\), then the probability that neither occurs is:

• \(P(A^c \cap B^c) = 1 - 0.7 = 0.3\)

This shows there's a 0.3 probability that neither A nor B takes place.

Exploring the Union of Events

The union of two events in probability considers the situation where either event A, event B, or both occur. This is akin to bringing together all elements from individual events A and B. Mathematically, the union is written as \(A \cup B\), and calculated using:

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

Subtracting the intersection \(P(A \cap B)\) is important to avoid double-counting the probability of both A and B occurring.
For instance, if both \(P(A)\) and \(P(B)\) are \(0.4\) with an intersection of \(0.1\), then

• \(P(A \cup B) = 0.4 + 0.4 - 0.1 = 0.7\)

Thus, there is a 0.7 probability that either A, B, or both will occur, illustrating how the union combines occurrences to show greater likelihood.