(a) It is noted that 8% of Kaplan students are left handed. If twenty (20) students are randomly selected, calculate the i. probability that none of them are left-handed, (2 marks) Solution: This question is based on the binomial distribution. Here we are given, n = 20 and p = 8% = 0.08 we have to calculate P(X=0) the formula for binomial distribution is given as below: P(X) = nCx * p^x * ( 1 – p )^(n-x) Now plug all values in this formula, P(X=20) = 20C0 * (0.08)^0 * (1 – 0.08)^(20-0) = 0.1887 Required probability = 0.1887 ii. probability that at most 2 are left-handed, (3 marks) here we have to find P(X≤2) we know, P(X≤2) = P(X=0) + P(X=1) + P(X=2) P(X=0) = 0.1887 ( already find in above question ) P(X=1) = 20C1 * (0.08)^1 * (1- 0.08)^(20 – 1) = 0.3282 P(X=2) = 20C2 * (0.08)^2 * (1-0.08)^(20-2) = 0.2711 P(X≤2) = 0.1887 + 0.3282 + 0.2711 = 0.7879 Required probability = 0.7879 iii. mean and standard deviation for the number of left-handed students (3 marks) solution: The mean is given as: Mean = n*p = 20*0.08 = 1.6 And standard deviation is given as: Standard deviation = sqrt(npq) = sqrt(20*0.08*(1-0.08)) = 1.2133 (b) Do you agree that "if two events are mutually exclusive then these two events will be independent"? Why? Provide one business-related example each, with explanation, for mutually exclusive and independent events. (7 marks) Solution: Yes, if two events are mutually exclusive then these two events will be independent because there are no same element exists in these two events. Example of mutually exclusive events: Risk of investing money in new business and risk of not investing money in new business is a example of mutually exclusive events. Example of independent events: Rolling a 2 on die and flipping a head on coin is a example of independent events.