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 how to solve the problem: 1. The value of x. Step 1: Define the total number of marbles and the probability of picking two black marbles. There are 7 black marbles and x red marbles. The total number of marbles is 7+x. The probability of picking two black marbles without replacement is the product of the probability of picking the first black marble and the probability of picking the second black marble (given the first was black): P(both black) = (7)/(7+x) × (6)/(6+x) Step 2: Set up the equation using the given probability. We are given that the probability of both marbles being black is (7)/(22). (7)/(7+x) × (6)/(6+x) = (7)/(22) (42)/((7+x)(6+x)) = (7)/(22) Step 3: Solve the equation for x. Divide both sides by 7: (6)/((7+x)(6+x)) = (1)/(22) Cross-multiply: 6 × 22 = (7+x)(6+x) 132 = 42 + 7x + 6x + x^2 132 = x^2 + 13x + 42 Rearrange into a quadratic equation: x^2 + 13x - 90 = 0 Step 4: Factor the quadratic equation and find x. We look for two numbers that multiply to -90 and add to 13. These numbers are 18 and -5. (x+18)(x-5) = 0 This gives two possible values for x: x = -18 or x = 5. Since the number of marbles cannot be negative, we choose the positive value. The value of x is 5. 2. The probability of picking one marble of each color. Step 1: Determine the total number of marbles with the value of x. With x=5: Number of black marbles = 7. Number of red marbles = 5. Total number of marbles = 7+5 = 12. Step 2: Calculate the probability of picking one black and one red marble. There are two mutually exclusive ways to pick one marble of each color: • Picking a black marble first, then a red marble (P(B then R)). • Picking a red marble first, then a black marble (P(R then B)). P(B then R) = (7)/(12) × (5)/(11) = (35)/(132) P(R then B) = (5)/(12) × (7)/(11) = (35)/(132) Step 3: Sum the probabilities and simplify. The total probability of picking one marble of each color is the sum of these two probabilities: P(one of each color) = P(B then R) + P(R then B) P(one of each color) = (35)/(132) + (35)/(132) = (70)/(132) Simplify the fraction by dividing the numerator and denominator by 2: P(one of each color) = (35)/(66) The probability of picking one marble of each color is (35)/(66). Send me the next one 📸