This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the questions. Question 1: Given: P(A) = (1)/(2), P(A B) = (2)/(9), P(A B) = (7)/(9). a) Calculate P(B) Step 1: Use the formula for the probability of the union of two events. P(A B) = P(A) + P(B) - P(A B) Step 2: Substitute the given values into the formula. (7)/(9) = (1)/(2) + P(B) - (2)/(9) Step 3: Solve for P(B). P(B) = (7)/(9) - (1)/(2) + (2)/(9) P(B) = (9)/(9) - (1)/(2) P(B) = 1 - (1)/(2) P(B) = (1)/(2) a) (1)/(2) b) Calculate P(A|B) Step 1: Use the formula for conditional probability. P(A|B) = (P(A B))/(P(B)) Step 2: Substitute the known values for P(A B) and P(B). P(A|B) = (2)/(9)(1)/(2) Step 3: Simplify the expression. P(A|B) = (2)/(9) × (2)/(1) P(A|B) = (4)/(9) b) (4)/(9) c) Calculate P(A' B) Step 1: Use the relationship between P(B), P(A B), and P(A' B). P(A' B) = P(B) - P(A B) Step 2: Substitute the known values for P(B) and P(A B). P(A' B) = (1)/(2) - (2)/(9) Step 3: Find a common denominator and subtract. P(A' B) = (9)/(18) - (4)/(18) P(A' B) = (5)/(18) c) (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 L_A) = P(F|L_A) × P(L_A) = 0.30 × 0.60 = 0.18 Step 2: Calculate the probability of a female candidate from List B. P(F L_B) = P(F|L_B) × P(L_B) = 0.40 × 0.40 = 0.16 Step 3: Sum these probabilities to find the total probability