Answer:
The probability that a customer will spend more than 15 minutes total in the bank, given that the customer has already waited over 10 minutes is 0.6065.
Explanation:
The random variable T is defined as the amount of time a customer spends in a bank.
The random variable T is exponentially distributed.
The probability density function of a an exponential random variable is:

The average time a customer spends in a bank is β = 10 minutes.
Then the parameter of the distribution is:

An exponential distribution has a memory-less property, i.e the future probabilities are not affected by any past data.
That is, P (X > s + x | X > s) = P (X > x)
So the probability that a customer will spend more than 15 minutes total in the bank, given that the customer has already waited over 10 minutes is:
P (X > 15 | X > 10) = P (X > 5)
![\int\limits^(\infty)_(5) {f(x)} \, dx =\int\limits^(\infty)_(5) {\lambda e^(-\lambda x)} \, dx\\=\int\limits^(\infty)_(5) {0.10 e^(-0.10 x)} \, dx\\=0.10\int\limits^(\infty)_(5) {e^(-0.10 x)} \, dx\\=0.10|(e^(-0.10 x))/(-0.10)|^(\infty)_(5)\\=[-e^(-0.10 * \infty)+e^(-0.10 * 5)]\\=0.6065](https://img.qammunity.org/2021/formulas/mathematics/college/bqn7gta1e4ldb8tal0b1rvt6odjgq84y14.png)
Thus, the probability that a customer will spend more than 15 minutes total in the bank, given that the customer has already waited over 10 minutes is 0.6065.