The reasoning is the same as for P(E), so we can write PO xp(- It might seem surprising that P(E)and P(n)differ, given that each conflict we are considering involves one eastbound and one northbound plane. If, however, 2N> NE (for example), then more northbound planes reach J per hour than do eastbound planes Thus, if equal numbers of northbound and eastbound planes face conflicts, the percentage of conflicts is lower for northbound planes passing through J than eastbound ones. And P(E)and P(N)reflect these percentages P(EE: The reader might be wondering: what is the difference between P(EE)and P(e)? The definitions of the events differ a bit: P(E)requires that a conflict be in progress when an eastbound plane reaches J; P(EE)requires some east/north conflict, but allows for the possibility that the conflict is already over(or has not yet begun) when the eastbound plane passes through J. Still, does this distinction really matter? Well, yes. Suppose that, when an eastbound plane arrives at J, there is a northbound plane six miles north of J. The two planes are not then in conflict. But consider the situation twelve seconds earlier, when the northbound plane was four miles north of J and the eastbound plane two miles west of it. The Pythagorean theorem reminds us that the two planes were v20=4.5 miles apart at that time(i. e. that they were in conflict, even though they no longer are) hrough a combination of geometry and calculus, one can reach a potent conclusion EE occurs if, at the moment the eastbound plane reaches J, there is a northbound aircraft within 10/2=7. 1 miles of the junction either way. (If the plane is between 5 and 7.1 miles away, then the conflict has already ended or not yet started. Because planes travelThe reasoning is the same as for P(E), so we can write: P(N) = 1 - exp(-
N) It might seem surprising that P(E) and P(N) differ, given that each conflict we are considering involves one eastbound and one northbound plane. If, however,
N >
E (for example), then more northbound planes reach J per hour than do eastbound planes. Thus, if equal numbers of northbound and eastbound planes face conflicts, the percentage of conflicts is lower for northbound planes passing through J than eastbound ones. And P(E) and P(N) reflect these percentages. P(EE): The reader might be wondering: what is the difference between P(EE) and P(E)? The definitions of the events differ a bit: P(E) requires that a conflict be in progress when an eastbound plane reaches J; P(EE) requires some east/north conflict, but allows for the possibility that the conflict is already over (or has not yet begun) when the eastbound plane passes through J. Still, does this distinction really matter? Well, yes. Suppose that, when an eastbound plane arrives at J, there is a northbound plane six miles north of J. The two planes are not then in conflict. But consider the situation twelve seconds earlier, when the northbound plane was four miles north of J and the eastbound plane two miles west of it. The Pythagorean theorem reminds us that the two planes were 20 = 4.5 miles apart at that time (i.e. that they were in conflict, even though they no longer are). Through a combination of geometry and calculus, one can reach a potent conclusion: EE occurs if, at the moment the eastbound plane reaches J, there is a northbound aircraft within 10/2 7.1 miles of the junction either way. (If the plane is between 5 and 7.1 miles away, then the conflict has already ended or not yet started.) Because planes travel