HOW LONG YOU HAVE TO WAIT FOR YOUR NEXT BUS OR EARTH-QUAKE?

   

     Whenever we wait for the city buses, they do not turn up for a long time or they come in groups of 3 or 4.  Why this happens?
  Say, buses in a particular route are scheduled to arrive at a 'stop' for every 12 minutes.  That is, 5 buses are expected in a bus-stop in an hour.  But they are bunched up or scattered away due to heavy traffic or other odd reasons.  But there is a statistical formula (exponential distribution) using which, we can calculate our 'chance' of getting  a bus in an interval of time.

      P = 1-eˆ(-rk)
P- probability or chance of occurrence of event in the interval of time 0 sec to k second.
    e- 2.71 ... Euler's constant.
    r - rate of occurrence of events

 In our case;
 k = 5 minutes
 r = 1/12:   one bus arrives for every 12 minutes. or 1/12 of a bus arrives at every minute.
Now p = 1- 2.71 ˆ[-5(1/12)]
            = 1-2.71 ˆ[-5/12]
            = 1- 1/ 2.71 ˆ5/12
            = 1-0.66  or 34 / 100 = 34%
Hence our chance of getting a bus in five minutes is only 34%

  Now we will calculate for 20 minutes.
p = 1 - 1/2.72ˆ(-20/12)
   = 1-0.19 = 0.81=81%
  in 20 minutes, our chances are bright.

     In this way, we can calculate the probabilities for the occurrence of earth quake, tsunami, burglary, epidemic etc.
    Suppose Tsunami arrives every 200 years in a place and the tsunami hits the shore today.  You can walk along the sea shore safely the very next day.  Because the arrival of Tsunami waves in the next 24 hours is extremely rare or the probability is nearly zero.
     Statistics and probability theory tries to predict the unpredictable.

FOOT NOTE;
     The mathematical constant 'e' appear in many places.  It is yet another use of it.  As the time increases, the chance increases rapidly.  Hence the name exponential distribution.
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