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, correcting Question 1 based on the provided image. Question 1: Given: P(A) = (1)/(2), P(A' B) = (1)/(4), P(A B) = (2)/(9). a) Calculate P(B) Step 1: Use the formula P(B) = P(A B) + P(A' B). P(B) = (2)/(9) + (1)/(4) Step 2: Find a common denominator and add the fractions. P(B) = (2 × 4)/(9 × 4) + (1 × 9)/(4 × 9) P(B) = (8)/(36) + (9)/(36) P(B) = (17)/(36) a) (17)/(36) 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)(17)/(36) Step 3: Simplify the expression. P(A|B) = (2)/(9) × (36)/(17) P(A|B) = (2 × 4)/(17) P(A|B) = (8)/(17) b) (8)/(17) c) Calculate P(A' B) This value is directly given in the problem statement. P(A' B) = (1)/(4) c) (1)/(4) 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) = 1 - 0.70 = 0.30 P(M|L_B) = 0.60 P(F|L_B) = 1 - 0.60 = 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 of a female candidate winning. P(F) = P(F L_A) + P(F L_B) = 0.18 + 0.16 = 0.34 a) 0.34 b) A female candidate wins given that she is from list B. This is asking for P(F|L_B). From the given information, P(F|L_B) = 1 - P(M|L_B) = 1 - 0.60 = 0.40. b) 0.40 3) Determine the two most popular candidates who win in the post parliamentary primaries. Step 1: Calculate the overall probability for each type of candidate. • Probability of a Male candidate from List A (M_A): P(M_A) = P(M|L_A) × P(L_A) = 0.70 × 0.60 = 0.42 • Probability of a Female candidate from List A (F_A): P(F_A) = P(F|L_A) × P(L_A) = 0.30 × 0.60 = 0.18 • Probability of a Male candidate from List B (M_B): P(M_B) = P(M|L_B) × P(L_B) = 0.60 × 0.40 = 0.24 • Probability of a Female candidate from List B (F_B): P(F_B) = P(F|L_B) × P(L_B) = 0.40 × 0.40 = 0.16 Step 2: Identify the two highest probabilities. The probabilities are 0.42, 0.18, 0.24, 0.16. The highest probability is 0.42, corresponding to the Male candidate from List A. The second highest probability is 0.24, corresponding to the Male candidate from List B. The two most popular candidates are the Male candidate from List A and the Male candidate from List B.