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 question 16: 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 the same as for the first draw. Tree Diagram: ` Start | |--- 1st Draw (G) --- P(G1) = 5/12 --- 2nd Draw (G) --- P(G2|G1) = 5/12 --- Outcome: GG, P(GG) = 5/12 * 5/12 = 25/144 | | | |--- 2nd Draw (B) --- P(B2|G1) = 7/12 --- Outcome: GB, P(GB) = 5/12 * 7/12 = 35/144 | |--- 1st Draw (B) --- P(B1) = 7/12 --- 2nd Draw (G) --- P(G2|B1) = 5/12 --- Outcome: BG, P(BG) = 7/12 * 5/12 = 35/144 | |--- 2nd Draw (B) --- P(B2|B1) = 7/12 --- Outcome: BB, P(BB) = 7/12 * 7/12 = 49/144 ` Possible Outcomes and their Probabilities: • P(GG) = P(G) × P(G) = (5)/(12) × (5)/(12) = (25)/(144) • P(GB) = P(G) × P(B) = (5)/(12) × (7)/(12) = (35)/(144) • P(BG) = P(B) × P(G) = (7)/(12) × (5)/(12) = (35)/(144) • P(BB) = P(B) × P(B) = (7)/(12) × (7)/(12) = (49)/(144) c) Find the probability of getting both blue balls. The probability of getting both blue balls is P(BB). P(BB) = 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) Alternatively, P(different colour) = 1 - P(same colour) = 1 - (37)/(72) = (72-37)/(72) = (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).