iNegotiate

I've spent a good part of my life negotiating with authority figures about how late I can return home. I belong to a certain social sect that thinks monsters and werewolves slink about the streets at night. The first time I did stay out late at night, I was quite disappointed by the conspicuous absence of vampires. 

Suppose the curfew-setters believe that the risk of something untoward happening at night is normally distributed, then after 9 PM, we can say that the probability of being a victim of crime increases steadily, reaching its peak at 1 AM. But even the criminals need to go home and get some sleep to be fresh and alert for the next day, so after 1 AM, the probability of crime reduces, returning to pre-9 PM levels at 5 AM. So parents shouldn't tell their kids to "return home by 12 or not return home at all." They should tell them to return home before midnight or after 3 AM, thereby avoiding travel during the peak crime hours. 

But behaviour usually doesn't follow this logic, so I'm forced to assume that the normal distribution idea doesn't appeal to most parents. In fact, considering how their impatience escalates with time, I surmise that they probably think that risk is uniformly distributed over the 9 PM to 6 AM interval. As time goes by, the total area under the curve increases and that explains their panic. Worry not. I have the perfect negotiation strategy. At all events, it is unlikely that the distribution of the probability of crime at night is a discrete distribution, because that would suggest that a crime can only be committed at specific points in time. A continuous distribution is far more plausible. However, in a continuous distribution, only intervals have positive probability. The probability of crime at any given point of time will be zero. Explain to your parents that while you understand that there is some risk spread over the time interval in question, if they think of your safety at a specific point in time, their fears are inconsistent with their beliefs. 

Let's assume that doesn't work either. As a final resort, I recommend negotiating for spending the night at a friend's place instead of coming home late. You don't actually have to spend the night at anyone's place. You push your friends to party till the wee hours of the morning and return home like a good kid just as the lamps are being lit and the prayers recited. Your parents will love your devotion to family life and your friends will think you are a party animal of sorts. If that isn't a win-win, I don't know what is.