The random variable G represents the number of goals scored by a school football team in a match. The distribution of G is recorded partially in the table below.
Number of goals g : 0 1 2 3 4
P(G=g) : 0.15 a 0.30 b 0.05
It is known that the probability of scoring exactly one goal is three times the probability of scoring exactly three goals.
Based on this information, answer the following.
a. (i) Find the values of a and b.
b. (ii) Calculate the probability that the team scores at most two goals in a match.
c. (iii) After inserting the values of a and b, show that the distribution obtained is a valid probability model.