Probability of nth event is between x and y

Question

I have a uniform distribution of age in the range $[a, b)$ with $a=42$ and $b=78$

So the probability that a person walks in a bank that is between $50$ and $70$ years of age would be $\frac{70-50}{78-42}$, right?

But I'm struggling with this problem where I'm supposed to find

1. The probability that 3 out of 7 customers in the bank are between $50$ and $70$ years of age
2. The probability that the $7th$ customer entering the bank is between $50$ and $70$ years of age
3. The probability that the $7th$ customer entering the bank is also the 3rd customer between $50$ and $70$ years of age

For $1.$, is it just $\frac{20}{\binom{7}{3}}$?

And for $2.$ and $3.$, I'm a bit lost so I need some help on them

Answer 1

Let's assume that the ages of different customers is independent (probably a safe assumption for the purposes of the problem). Let $p = \frac{70 - 50}{78 - 42}$.

1. The probability that exactly 3 of the 7 are ages 50-70 is $\binom{3}{7}p^3(1-p)^4$. You can view this as a binomial distribution over 7 "coins", each with probability $p$ of being heads.

2. The probability of the 7th customer being in the age range is the same as the probability of any other customer being in that age range, which is $p$.

3. Here are some hints. If the 7th customer is the 3rd customer in the age range, then among the 6 preceding customers there must be 2 customers in that age range. What is the probability that exactly 2 of the first 6 customers are in the age range? We already know the probability that the 7th customer is in the age range, and that the age of the 7th customer is independent of the first 6. So what is the probability that 2 out of the first 6 are in the range AND that the 7th is in the range?