Given: P(A) = 1/2, P(A B) = 2/9, P(A B) = 7/9.

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Question 1:
Given: $P(A) = \frac{1}{2}$ , $P(A \cap B) = \frac{2}{9}$ , $P(A \cup B) = \frac{7}{9}$ .

a) Calculate $P(B)$

Use the formula for the probability of the union of two events.
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Substitute the given values into the formula.
$\frac{7}{9} = \frac{1}{2} + P(B) - \frac{2}{9}$
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 $P(A|B)$
Use the formula for conditional probability.
$P(A|B) = \frac{P(A \cap B)}{P(B)}$
Substitute the known values for $P(A \cap B)$ and $P(B)$ .
$P(A|B) = \frac{\frac{2}{9}}{\frac{1}{2}}$
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 $P(A' \cap B)$
Use the relationship between $P(B)$ , $P(A \cap B)$ , and $P(A' \cap B)$ .
$P(A' \cap B) = P(B) - P(A \cap B)$
Substitute the known values for $P(B)$ and $P(A \cap B)$ .
$P(A' \cap B) = \frac{1}{2} - \frac{2}{9}$
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.
Calculate the probability of a female candidate from List A.
$P(F \cap L_A) = P(F|L_A) \times P(L_A) = 0.30 \times 0.60 = 0.18$
Calculate the probability of a female candidate from List B.
$P(F \cap L_B) = P(F|L_B) \times P(L_B) = 0.40 \times 0.40 = 0.16$
Sum these probabilities to find the total probability

Given: P(A) = (1)/(2), P(A B) = (2)/(9), P(A B) = (7)/(9).

Step-by-step solution

Use the formula for the probability of the union of two events.

Substitute the given values into the formula.

Solve for $P(B)$ .

Use the formula for conditional probability.

Substitute the known values for $P(A \cap B)$ and $P(B)$ .

Simplify the expression.

Use the relationship between $P(B)$ , $P(A \cap B)$ , and $P(A' \cap B)$ .

Substitute the known values for $P(B)$ and $P(A \cap B)$ .

Find a common denominator and subtract.

Calculate the probability of a female candidate from List A.

Calculate the probability of a female candidate from List B.

Sum these probabilities to find the total probability

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