The three time estimates for the activities of the network shown in the given figure are shown above their arrows. The earliest expected time for the event 4, is A. 19 B. 14 C. 24 D. None of these

The correct answer is $\boxed{\text{B) 14}}$.

The earliest expected time for an event is the earliest time that the event can occur, given the earliest expected times of the events that must precede it. In this case, event 4 can only occur after events 1, 2, and 3 have occurred. The earliest expected times of events 1, 2, and 3 are 3, 5, and 7, respectively. Therefore, the earliest expected time for event 4 is $3 + 5 + 7 = \boxed{14}$.

Here is a step-by-step explanation of how to calculate the earliest expected time for event 4:

1. Calculate the earliest expected time for each event that must precede event 4. In this case, the events that must precede event 4 are events 1, 2, and 3. The earliest expected times of events 1, 2, and 3 are 3, 5, and 7, respectively.
2. Add the earliest expected times of the events that must precede event 4. This gives us $3 + 5 + 7 = 14$.
3. This is the earliest expected time for event 4.