The probability of finding a given card after exactly n tries is (12/13)n-1(1/13) = (1/12) (12/13)n, so the expected number of steps before finding a given card is
<steps>=(1/12) Sn=1¥ [n (12/13)n]
To evaluate this sum, recall that for x<>1
Sn=0N [xn] = (1-xN)/(1-x)
so that Sn=1N [xn] = (x-xN)/(1-x)
and Sn=2N [xn] = (x2-xN)/(1-x)
For x<1 we can take the limit as N goes to ¥ to get
Sn=0¥ [xn] = 1/(1-x)
Sn=1¥ [xn] = x/(1-x)
and Sn=2¥ [xn] = x2/(1-x)
Differentiate the last of these to get
Sn=2¥ [n xn-1] = [2 x (1-x)+x2]/(1-x)2
Substituting n'=n-1 gives Sn'=1¥ [(n'+1) xn'] = [2 x (1-x)+x2]/(1-x)2
which simplifies to
Sn=1¥ [n xn] = x/(1-x)2
Hence, as should be expected, <steps> = 13.