The uploaded image contains a math quiz focusing on the "Probability of Events," specifically analyzing the relationship between two events, $A$ and $B$, with probabilities $P(A) = \frac{1}{2}$, $P(A \cup B) = \frac{2}{3}$, and $P(A \cap B) = \frac{1}{6}$. The task is to find $P(B)$. Here’s how we can solve it step by step:

Solution:

The formula for the union of two events is: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Substitute the given values:

$\frac{2}{3} = \frac{1}{2} + P(B) - \frac{1}{6}$

Simplify:

First, find a common denominator for the fractions:

• $\frac{1}{2} = \frac{3}{6}$,
• $\frac{2}{3} = \frac{4}{6}$.

Thus, the equation becomes: $\frac{4}{6} = \frac{3}{6} + P(B) - \frac{1}{6}.$

Combine terms: $\frac{4}{6} = \frac{2}{6} + P(B).$

Solve for $P(B)$: $P(B) = \frac{4}{6} - \frac{2}{6} = \frac{2}{6} = \frac{1}{3}.$

Final Answer:

$P(B) = \frac{1}{3}.$

Would you like a detailed explanation of the derivation or have other related questions? Here are five related questions you might explore:

1. How do you calculate $P(A \cap B^c)$ using these probabilities?
2. How does the independence of events $A$ and $B$ affect $P(A \cup B)$?
3. Can you derive $P(B^c)$ from $P(B)$?
4. How do you check if $A$ and $B$ are mutually exclusive using the provided values?
5. What is the probability of $A^c \cap B$, and how do you calculate it?

Tip:

Always remember that $P(A \cup B)$ can be rearranged to find any missing component if the values for $P(A)$, $P(B)$, and $P(A \cap B)$ are provided.