Here are the solutions to the questions. Question 1: Given: $P(A) = \frac{1}{2}$, $P(A \cap B) = \frac{2}{9}$, $P(A \cup B) = \frac{7}{9}$. a) Calculate $P(B)$ Step 1: Use the formula for the probability of the union of two events. $$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$ Step 2: Substitute the given values into the formula. $$ \frac{7}{9} = \frac{1}{2} + P(B) - \frac{2}{9} $$ Step 3: Solve for $P(B)$. $$ P(B) = \frac{7}{9} - \frac{1}{2} + \frac{2}{9} $$ $$ P(B) = \frac{9}{9} - \frac{1}{2} $$ $$ P(B) = 1 - \frac{1}{2} $$ $$ P(B) = \frac{1}{2} $$ a) $\boxed{\frac{1}{2}}$ b) Calculate $P(A|B)$ Step 1: Use the formula for conditional probability. $$ P(A|B) = \frac{P(A \cap B)}{P(B)} $$ Step 2: Substitute the known values for $P(A \cap B)$ and $P(B)$. $$ P(A|B) = \frac{\frac{2}{9}}{\frac{1}{2}} $$ Step 3: Simplify the expression. $$ P(A|B) = \frac{2}{9} \times \frac{2}{1} $$ $$ P(A|B) = \frac{4}{9} $$ b) $\boxed{\frac{4}{9}}$ c) Calculate $P(A' \cap B)$ Step 1: Use the relationship between $P(B)$, $P(A \cap B)$, and $P(A' \cap B)$. $$ P(A' \cap B) = P(B) - P(A \cap B) $$ Step 2: Substitute the known values for $P(B)$ and $P(A \cap B)$. $$ P(A' \cap B) = \frac{1}{2} - \frac{2}{9} $$ Step 3: Find a common denominator and subtract. $$ P(A' \cap B) = \frac{9}{18} - \frac{4}{18} $$ $$ P(A' \cap B) = \frac{5}{18} $$ c) $\boxed{\frac{5}{18}}$ Question 2: Let $L_A$ be the event that a voter preferred List A, and $L_B$ be the event that a voter preferred List B. Let $M$ be the event that a voter preferred a male candidate, and $F$ be the event that a voter preferred a female candidate. Given probabilities: $P(L_A) = 0.60$ $P(L_B) = 0.40$ $P(M|L_A) = 0.70$ $P(F|L_A) = 0.30$ $P(M|L_B) = 0.60$ $P(F|L_B) = 0.40$ 1) Draw a tree diagram to illustrate this information. ` Start | |-- P(L_A) = 0.60 --|-- P(M|L_A) = 0.70 --> Male from List A (M_A) | |-- P(F|L_A) = 0.30 --> Female from List A (F_A) | |-- P(L_B) = 0.40 --|-- P(M|L_B) = 0.60 --> Male from List B (M_B) |-- P(F|L_B) = 0.40 --> Female from List B (F_B) ` 2) Calculate the probability that: a) A female candidate wins. Step 1: Calculate the probability of a female candidate from List A. $$ P(F \cap L_A) = P(F|L_A) \times P(L_A) = 0.30 \times 0.60 = 0.18 $$ Step 2: Calculate the probability of a female candidate from List B. $$ P(F \cap L_B) = P(F|L_B) \times P(L_B) = 0.40 \times 0.40 = 0.16 $$ Step 3: Sum these probabilities to find the total probability