Assume that your are the next to last customer in a bank bef

Assume that your are the next to last customer in a bank befoer closing time. There are three tellers and each one is serving one customer, there are two customers waiting to be served and you are first of the these. Assume that all service times are indepedent and exponenetially distributed witha mean of five minutes. What is the probability that you will be the last person to finish your transaction?

Solution

let, V = max( X , Y, Z )...
p [ V <= t ] = P [ X <= t ] . P [ Y <= t ] . P [ Z <= t ] = (1 - e ^ ( - x/5 ) ) ^3.......

fV(t) = 3*( (1 - e ^ ( - x/5 ) ) ^2 * ( e^( - x/5) /5 ) ...

so, this is the distribution of maiximum of exponential!

now, exponential is memoryless distribution!
so, it doesnot remember who is waiting at what position!

so, each person has equal chanc eof finishing last!
P [ i will be the last person ] = 1/3......