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.
ScanSolve AI Answer
Here's the solution for question 5: a) Find the number of z marbles in the bag. Step 1: Define the number of marbles and set up ratios. Let N_x, N_y, N_z be the number of marbles of type x, y, z respectively. The total number of marbles is N_x + N_y + N_z = 27. The given ratios are N_x : N_y = 1 : 2 and N_y : N_z = 1 : 3. From N_x : N_y = 1 : 2, we have N_y = 2N_x, which means N_x = (1)/(2)N_y. From N_y : N_z = 1 : 3, we have N_z = 3N_y. Step 2: Express all marble counts in terms of N_y. Substitute N_x = (1)/(2)N_y and N_z = 3N_y into the total marbles equation: (1)/(2)N_y + N_y + 3N_y = 27 Step 3: Solve for N_y and then N_z. Combine the terms: ((1)/(2) + 1 + 3)N_y = 27 ((1)/(2) + (2)/(2) + (6)/(2))N_y = 27 (9)/(2)N_y = 27 Multiply both sides by (2)/(9): N_y = 27 × (2)/(9) N_y = 3 × 2 N_y = 6 Now, calculate the number of z marbles using N_z = 3N_y: N_z = 3 × 6 N_z = 18 The number of z marbles in the bag is 18. b) Calculate the probability of selecting two of the y marbles at random from the bag one after the other without replacement. Step 1: Identify the number of y marbles and total marbles. From part (a), the number of y marbles is N_y = 6. The total number of marbles in the bag is 27. Step 2: Calculate the probability of selecting the first y marble. The probability of selecting a y marble on the first draw is the number of y marbles divided by the total number of marbles. P(1st y) = (6)/(27) Step 3: Calculate the probability of selecting the second y marble without replacement. After one y marble is selected and not replaced, there are 6 - 1 = 5 y marbles remaining. The total number of marbles remaining in the bag is 27 - 1 = 26. The probability of selecting a second y marble is: P(2nd y | 1st y) = (5)/(26) Step 4: Calculate the total probability. The probability of selecting two y marbles one after the other without replacement is the product of the probabilities from Step 2 and Step 3. P(two y) = P(1st y) × P(2nd y | 1st y) P(two y) = (6)/(27) × (5)/(26) Simplify the fractions: P(two y) = (2)/(9) × (5)/(26) P(two y) = (2 × 5)/(9 × 26) P(two y) = (10)/(234) Further simplify by dividing the numerator and denominator by 2: P(two y) = (5)/(117) The probability is (5)/(117). 3 done, 2 left today. You're making progress.