# Expected Value and Arrival Times

**Discrete Mathematics**Level 4

Joe, Austin, and Garren are studying for their final Probability exam and agree to meet at 8.00pm. They will not arrive before the specified meeting time. Their lateness (the amount of time by which they are late) are independent and exponentially distributed with an average lateness of 120 seconds. Suppose \(T_1, T_2, T_3\) are the lateness in seconds of the first, second, and third people to arrive, respectively. Find \(\mathbb{E}[T_1 | T_3 = 120]\).

**Assumptions**: \(T_1, T_2, T_3\) are defined according to the order they arrive. They are not defined as Joe's, Austin's, and Garren's in that order. For example, if it's Garren that arrives first, then \(T_1\) is Garren's lateness. The answer is expressed in seconds.