Suppose that the probability that a given customer entering a bank is between 50 and 70 years of age is 5/9. a.) On a given day, compute the probability that the 7th customer entering the bank is also the 3rd customer who is between 50 and 70 years of age. b.) With 7 people present in the bank, compute the probability that 3 of them are between 50 and 70 years of age.
For the question, above how should I apply Poisson Distribution to figure out the answer, for this question is stated to be a Poisson Process?
No, you don't use a Poisson distribution, because this isn't talking about the number of customers in a given time interval. This question is assuming that each customer has probability $5/9$ of being from $50$ to $70$, independent of the others (that should be stated better - it also may not be true in real life, as customers often come together with others of similar age, and also the age distribution is likely to depend on time of day). It's also assuming that at least $7$ customers do enter the bank on that day. I would expect that to have quite a high probability, but not exactly $1$ (and certainly not $1$ if it's a Poisson process).
For (a) you want the probability that there are $2$ people $50$ to $70$ among the first $6$ customers (use Binomial for that) and then the $7$'th customer is also $50$ to $70$.