(1)  First, I calculate the probability in (a) as

         P = normalcfd(20, 30, 32.3, 9.7) = 0.3039.

     I consider the age here as the continuous variable (time), not necessary
     with integer values: it changes from 20 = 20+0 to 30 = 30-0.


     And I got the value of probability 0.3039, different from yours value of 0.23.




(2)  This probability that I got (of 0.3039) is not only probability that the next customer will be at his (or her) twenties:

     it is the probability that ANY random customer's age is in this interval.


     Here they slightly confuse you (intently) with their formulations, 
     but you should understand, where do they confuse you - it is the rule/(the part) of the game.




(3)  Part (b) of the problem relates to Binomial distribution with n= 3 (the number of trials) 
     and k= 2 (the number of success); p = 0.3039, q = 1-p = 1-0.3039 = 0.6961.


     Therefore,  P =  =  = 0.192  (rounded).


     Here there is no limitations for the binomial distribution formula: we do not use the normal distribution 
     as an approximation to the binomial distribution.


     We use here the binomial distribution formula as is, for its own correct purposes.