Suppose A1, A2, and A3 are three events in a sample space. Let us consider the sums S1 = P(A1) + P(A2) + P(A3), S2 = P(A1A2) + P(A1A3) + P(A2A3), S3 = P(A1A2A3).

Express mathematically, in terms of S1, S2, and S3, the probability for the event that

(i) exactly one of the Ai occurs;

(ii) exactly two of the Ai occur;

(iii) none of the Ai occur.

Application: Among the customers of an insurance company, the percentages of those who have home insurance, liability insurance and car insurance are respectively, 35%, 15%, and 40%. Suppose 8% have both home and liability insurances, 25% have both home and car insurances, and 10% have both liability and car insurances. Finally, 3% of the customers have all three types of insurance.

For i = 0, 1, 2, calculate the percentage of persons who have exactly i types of insurance with the company.