Let's solve the questions given in the image.

Given:

• $P(A) = \frac{1}{4}$
• $P(B) = \frac{1}{5}$
• $P(A \cap B) = \frac{1}{7}$

Questions:

1. Find $P(A \cup B)$:

   The formula for the union of two events $A$ and $B$ is: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ Plugging in the values: $P(A \cup B) = \frac{1}{4} + \frac{1}{5} - \frac{1}{7}$ We need a common denominator to perform the addition and subtraction. The least common multiple of 4, 5, and 7 is 140.

   $P(A \cup B) = \frac{35}{140} + \frac{28}{140} - \frac{20}{140} = \frac{43}{140}$

   So, $P(A \cup B) = \frac{43}{140}$.

2. Find $P(A | B)$:

   The formula for the conditional probability $P(A | B)$ is: $P(A | B) = \frac{P(A \cap B)}{P(B)}$ Substituting the values: $P(A | B) = \frac{\frac{1}{7}}{\frac{1}{5}} = \frac{1}{7} \times \frac{5}{1} = \frac{5}{7}$

   Therefore, $P(A | B) = \frac{5}{7}$.

3. Find $P(B | A^c)$:

   Here, $A^c$ denotes the complement of $A$. The formula for $P(B | A^c)$ is: $P(B | A^c) = \frac{P(B \cap A^c)}{P(A^c)}$ We know that: $P(A^c) = 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4}$

   To find $P(B \cap A^c)$, we use: $P(B) = P(B \cap A) + P(B \cap A^c)$ Rearranging, we get: $P(B \cap A^c) = P(B) - P(A \cap B) = \frac{1}{5} - \frac{1}{7}$

   Finding a common denominator (35): $P(B \cap A^c) = \frac{7}{35} - \frac{5}{35} = \frac{2}{35}$

   Thus, $P(B | A^c) = \frac{\frac{2}{35}}{\frac{3}{4}} = \frac{2}{35} \times \frac{4}{3} = \frac{8}{105}$

   So, $P(B | A^c) = \frac{8}{105}$.

Summary of Answers:

• a. $P(A \cup B) = \frac{43}{140}$
• b. $P(A | B) = \frac{5}{7}$
• c. $P(B | A^c) = \frac{8}{105}$

Would you like more detailed explanations for each step, or do you have any specific questions?

Further Related Questions:

1. How do you calculate $P(A^c \cup B)$?
2. What would $P(B^c | A)$ represent, and how is it calculated?
3. How would the answer change if $P(A \cap B)$ were different?
4. Can you explain the difference between $P(A \cap B)$ and $P(A | B)$?
5. What is the formula for conditional probability when events are independent?

Tip:

When solving probability problems, remember to always check if events are independent, as it can simplify calculations.