EE: the chance that a given eastbound plane that passes through J is at any time conflict with a northbound plane that passes through J P(EI To find P(E), we note that the conflict occurs if, at the time the eastbound plane reaches J, there is a northbound plane within five miles of J. If E* is the complement of E, then E* requires that there be no northbound plane within five miles of the junction. It is easier to find P(es)than P(e), so we will do so and then invoke the rule p(e)=1-p(es) We arent told anything about planes that are not at the junction, so how can we determine whether an aircraft is within five miles of J? We can exploit the clue that planes travel at 600 miles per hour(which works out to ten miles per minute, or one mile every six seconds ). Suppose that a plane is north of j and within five miles of it. Then the plane must have passed through J within the last thirty seconds. Similarly, if a northbound plane is still south of J but less than five miles away, it will reach J within the next thirty seconds Thus, if an eastbound plane reaches J at time t, there will be a conflict at t if a northbound plane passes through J between t-0.5(in minutes)and t +0.5. And there will be no conflict if no northbound plane reaches t over the interval(t-0.5, t+0.5) e can therefore write P(E*)= P(no northbound arrivals at J over(t-0.5, t+0. 5))=exp(-AN) and thus that P(E)=1-exp(-AN) P(N:EE: the chance that a given eastbound plane that passes through J is at any time in conflict with a northbound plane that passes through J. P(E) To find P(E), we note that the conflict occurs if, at the time the eastbound plane reaches J, there is a northbound plane within five miles of J. If E* is the complement of E, then E* requires that there be no northbound plane within five miles of the junction. It is easier to find P(E*) than P(E), so we will do so and then invoke the rule P(E) = 1 - P(E*). We aren’t told anything about planes that are not at the junction, so how can we determine whether an aircraft is within five miles of J? We can exploit the clue that planes travel at 600 miles per hour (which works out to ten miles per minute, or one mile every six seconds). Suppose that a plane is north of J and within five miles of it. Then, the plane must have passed through J within the last thirty seconds. Similarly, if a northbound plane is still south of J but less than five miles away, it will reach J within the next thirty seconds. Thus, if an eastbound plane reaches J at time t, there will be a conflict at t if a northbound plane passes through J between t-0.5 (in minutes) and t + 0.5. And there will be no conflict if no northbound plane reaches t over the interval (t-0.5,t+0.5). We can therefore write: P(E*) = P(no northbound arrivals at J over (t-0.5,t+0.5)) = exp(-
N) and thus that P(E) = 1 - exp(-
N) P(N):