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: Use Addition Rule for Probability

The addition rule of probability states that \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). We know \(P(A) = 0.4\), \(P(B) = 0.4\), and \(P(A \cup B) = 0.7\). Substitute these values into the equation to find \(P(A \cap B)\).

Step 2: Solve for Intersection Probability

Substituting the values into the equation gives:\[0.7 = 0.4 + 0.4 - P(A \cap B)\]Simplifying this equation:\[0.7 = 0.8 - P(A \cap B)\]\[P(A \cap B) = 0.8 - 0.7\]\[P(A \cap B) = 0.1\].

Step 3: Use Complement Rule for Intersection of Complements

Using the complement rule, we know that:\[P(A^c \cap B^c) = 1 - P(A \cup B)\].Substituting the value of \(P(A \cup B)\), we have:\[P(A^c \cap B^c) = 1 - 0.7 = 0.3\].

Key Concepts

Addition Rule for Probability

In probability theory, understanding how to work with different events is crucial. The Addition Rule for Probability is a fundamental tool when dealing with the probability of either one event or another happening.

The rule is primarily expressed as follows:

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

Here, \(P(A \cup B)\) is the probability that either event \(A\) or event \(B\), or both, will occur.
\(P(A)\) and \(P(B)\) are the probabilities of events \(A\) and \(B\) occurring, respectively.
Finally, \(P(A \cap B)\) is the probability that both events \(A\) and \(B\) will occur simultaneously.

For example, if we know that \(P(A) = 0.4\), \(P(B) = 0.4\), and \(P(A \cup B) = 0.7\), we can plug these into our formula to find \(P(A \cap B)\).
**Understanding this rule allows us to break down seemingly complex probability problems**
into simpler parts, solving them with ease and accuracy.

Intersection Probability

The intersection probability involves finding the probability of two events occurring together.

Mathematically, the intersection of events \(A\) and \(B\) is denoted by \(P(A \cap B)\).

**To determine \(P(A \cap B)\), we can use the Addition Rule for Probability**:

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

This formula is invaluable when dealing with overlapping events.
For example, if events are not mutually exclusive, they can happen at the same time, making the intersection probability non-zero.

In our exercise, substituting the given values gives us: \[ P(A \cap B) = 0.4 + 0.4 - 0.7 = 0.1 \].

Therefore, there is a 0.1 probability that both events \(A\) and \(B\) occur together. **Recognizing and calculating intersection probabilities helps manage scenarios**
where events might not be independent of each other.

Complement Rule

The Complement Rule is an essential concept to grasp when dealing with probability, particularly in determining non-occurrence.

The complement of an event is the probability that the event does not happen and is denoted by \(A^c\).

According to the complement rule:

• \[ P(A^c) = 1 - P(A) \]

**Understanding this rule can simplify complex probability scenarios**.

When applying this to the intersection of complements, the formula becomes:

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

By determining \(P(A^c \cap B^c)\), we are effectively finding the probability that neither \(A\) nor \(B\) occur.

With the given value \(P(A \cup B) = 0.7\), then\( P(A^c \cap B^c) = 1 - 0.7 = 0.3 \).

This shows a 0.3 probability that neither event \(A\) nor \(B\) takes place.**Utilizing the complement rule is especially beneficial**
when it is easier to calculate the probability of the complement than the probability of an event directly.