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 are the solutions to the problems: 4. (b) Find the probability that none of them will pass the examination. Step 1: Calculate the probability of each person failing the examination. P(Mercy fails) = 1 - P(Mercy passes) = 1 - (2)/(3) = (1)/(3) P(John fails) = 1 - P(John passes) = 1 - (5)/(8) = (3)/(8) P(Alice fails) = 1 - P(Alice passes) = 1 - (3)/(4) = (1)/(4) Step 2: Multiply the probabilities of all three failing, as these are independent events. P(none pass) = P(Mercy fails) × P(John fails) × P(Alice fails) P(none pass) = (1)/(3) × (3)/(8) × (1)/(4) Step 3: Simplify the expression. P(none pass) = (1 × 3 × 1)/(3 × 8 × 4) = (3)/(96) P(none pass) = (1)/(32) 4. (c) Find the probability that only Mercy and John will pass the examination. Step 1: Identify the probabilities for Mercy passing, John passing, and Alice failing. P(Mercy passes) = (2)/(3) P(John passes) = (5)/(8) P(Alice fails) = (1)/(4) (from part b) Step 2: Multiply these probabilities together. P(only Mercy and John pass) = P(Mercy passes) × P(John passes) × P(Alice fails) P(only Mercy and John pass) = (2)/(3) × (5)/(8) × (1)/(4) Step 3: Simplify the expression. P(only Mercy and John pass) = (2 × 5 × 1)/(3 × 8 × 4) = (10)/(96) P(only Mercy and John pass) = (5)/(48) 5. (a) In the diagram, MK is a tangent to the circle at P, QRP = 50^ and SPK = 62^. Find QPS. Step 1: Use the Alternate Segment Theorem. The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. For tangent PK and chord PS: PQS = SPK = 62^ Step 2: Use the property that angles subtended by the same arc at the circumference are equal. Arc PQ subtends PRQ and PSQ. PSQ = PRQ = 50^ Step 3: Consider the triangle PQS. The sum of angles in a triangle is 180^. QPS + PQS + PSQ = 180^ Step 4: Substitute the known angles and solve for QPS. QPS + 62^ + 50^ = 180^ QPS + 112^ = 180^ QPS = 180^ - 112^ QPS = 68^ That's 2 down. 3 left today — send the next one.