Probability - The Addition Rule - Free Printable

Problem Analysis:

The problem involves understanding and applying the Addition Rule in probability. The Addition Rule is used to calculate the probability of the union of two events, i.e., the probability that at least one of the two events occurs. The formula for the Addition Rule is:

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

Where:
- \( P(A) \) is the probability of event \( A \).
- \( P(B) \) is the probability of event \( B \).
- \( P(A \cap B) \) is the probability of both events \( A \) and \( B \) occurring simultaneously.

The Venn diagram provided shows the probabilities of different regions:
- \( P(A \cap B^c) = 0.5 \) (probability of \( A \) but not \( B \)).
- \( P(A^c \cap B) = 0.1 \) (probability of \( B \) but not \( A \)).
- \( P(A \cap B) = 0.3 \) (probability of both \( A \) and \( B \)).
- \( P(A^c \cap B^c) = 0.1 \) (probability of neither \( A \) nor \( B \)).

We will use this information to solve the given questions.

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Solution to Each Part:

#### (a) What is the probability of \( A \)?
The probability of event \( A \) is the sum of the probabilities of the regions where \( A \) occurs:
- \( P(A \cap B^c) = 0.5 \) (probability of \( A \) but not \( B \)).
- \( P(A \cap B) = 0.3 \) (probability of both \( A \) and \( B \)).

Thus:
\[
P(A) = P(A \cap B^c) + P(A \cap B) = 0.5 + 0.3 = 0.8
\]

#### (b) What is the probability of \( A \) or \( B \) (or both)?
The probability of \( A \) or \( B \) (or both) is the union of events \( A \) and \( B \), denoted as \( P(A \cup B) \). Using the Addition Rule:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

First, we need to find \( P(B) \):
- \( P(A^c \cap B) = 0.1 \) (probability of \( B \) but not \( A \)).
- \( P(A \cap B) = 0.3 \) (probability of both \( A \) and \( B \)).

Thus:
\[
P(B) = P(A^c \cap B) + P(A \cap B) = 0.1 + 0.3 = 0.4
\]

Now, using the Addition Rule:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.8 + 0.4 - 0.3 = 0.9
\]

#### (c) What is the probability of \( A \) and \( B \)?
From the Venn diagram, the probability of both \( A \) and \( B \) occurring is directly given:
\[
P(A \cap B) = 0.3
\]

#### (d) What is the probability of \( A \) but not \( B \)?
The probability of \( A \) but not \( B \) is the region \( A \cap B^c \), which is directly given in the Venn diagram:
\[
P(A \cap B^c) = 0.5
\]

#### (e) What is the probability of \( A \) or \( B \) but not both?
The probability of \( A \) or \( B \) but not both is the sum of the probabilities of the regions where either \( A \) or \( B \) occurs, but not both:
- \( P(A \cap B^c) = 0.5 \) (probability of \( A \) but not \( B \)).
- \( P(A^c \cap B) = 0.1 \) (probability of \( B \) but not \( A \)).

Thus:
\[
P(\text{either } A \text{ or } B \text{ but not both}) = P(A \cap B^c) + P(A^c \cap B) = 0.5 + 0.1 = 0.6
\]

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Final Answers:

\[
\boxed{
\begin{array}{l}
\text{(a)} \; 0.8 \\
\text{(b)} \; 0.9 \\
\text{(c)} \; 0.3 \\
\text{(d)} \; 0.5 \\
\text{(e)} \; 0.6 \\
\end{array}
}
\]