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's the solution to the question: Given: Number of green balls (N_G) = 5 Number of blue balls (N_B) = 7 Total number of balls (N_T) = N_G + N_B = 5 + 7 = 12 Two balls are drawn randomly one after another with replacement. a) If A and B are two independent events, write the formula of the multiplication law of probability. For two independent events A and B, the multiplication law of probability states: P(A and B) = P(A B) = P(A) × P(B) b) Show the probability of all the possible outcomes in a tree diagram. The probability of drawing a green ball is P(G) = (5)/(12). The probability of drawing a blue ball is P(B) = (7)/(12). Since the draws are with replacement, the probabilities for the second draw are independent of the first. Tree Diagram Outcomes and Probabilities: • First draw Green (G_1), Second draw Green (G_2): P(G_1 and G_2) = P(G_1) × P(G_2) = (5)/(12) × (5)/(12) = (25)/(144) • First draw Green (G_1), Second draw Blue (B_2): P(G_1 and B_2) = P(G_1) × P(B_2) = (5)/(12) × (7)/(12) = (35)/(144) • First draw Blue (B_1), Second draw Green (G_2): P(B_1 and G_2) = P(B_1) × P(G_2) = (7)/(12) × (5)/(12) = (35)/(144) • First draw Blue (B_1), Second draw Blue (B_2): P(B_1 and B_2) = P(B_1) × P(B_2) = (7)/(12) × (7)/(12) = (49)/(144) c) Find the probability of getting both blue balls. The probability of getting both blue balls is P(B_1 and B_2). P(both blue) = P(B) × P(B) = (7)/(12) × (7)/(12) = (49)/(144) The probability of getting both blue balls is (49)/(144). d) Find the ratio of the probability of getting balls of the same colour to the different colour. Step 1: Calculate the probability of getting balls of the same colour. This occurs if both balls are green (GG) or both are blue (BB). P(same colour) = P(GG) + P(BB) P(same colour) = (25)/(144) + (49)/(144) = (25+49)/(144) = (74)/(144) = (37)/(72) Step 2: Calculate the probability of getting balls of different colour. This occurs if the first is green and the second is blue (GB), or the first is blue and the second is green (BG). P(different colour) = P(GB) + P(BG) P(different colour) = (35)/(144) + (35)/(144) = (35+35)/(144) = (70)/(144) = (35)/(72) Step 3: Find the ratio of P(same colour) to P(different colour). Ratio = P(same colour)P(different colour) = (37)/(72)(35)/(72) = (37)/(35) The ratio of the probability of getting balls of the same colour to the different colour is (37)/(35).