Pill puzzle

The problem becomes very easy to solve once a binary variable Xk defined as Xk = 1, if the kth half pill remains inside the jar after all the whole pills are removed. The kth half pill is defined as the result of the breaking of the kth whole pill being removed from the jar. Xk = 1 if out of the n − k + 1 pills (n − k whole pills + kth half pill), the one half pill is removed at the very end. This occurs with probability 1/(n − k + 1). The expected value is then given by, E(X1) + E(X2) + ... + E(Xn). Since E(Xk) = P(Xk = 1) = 1/(n − k + 1), the sought expected value is 1/n + 1/(n − 1) + 1/(n − 2) + ... + 1 = Hn (the nth harmonic number). ==References==