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 probability problems: Part (a) Let X be the number of bags found to be less than 50 kg. This is a binomial distribution problem, where n=8 (number of trials) and p=0.12 (probability of success, i.e., a bag being less than 50 kg). The probability mass function is P(X=k) = nk p^k (1-p)^n-k. a) (i) Exactly half Step 1: Identify the parameters for the binomial distribution. n = 8 p = 0.12 1-p = 0.88 We need to find the probability that exactly half of the bags are less than 50 kg, so k = (8)/(2) = 4. Step 2: Apply the binomial probability formula. P(X=4) = 84 (0.12)^4 (0.88)^8-4 P(X=4) = (8!)/(4!(8-4)!) (0.12)^4 (0.88)^4 P(X=4) = (8 × 7 × 6 × 5)/(4 × 3 × 2 × 1)