You need to use == rather than = when using Solve (and similar functions). However there is no n that makes the sum exactly equal to 1/2.
Here's a brute-force method: Calculate a bunch of values for n and then take the smallest n that results in a probability of at least 0.5:
data = Table[{n, N[Sum[(-1)^(k - 1)*Binomial[71, k]*(1 - (71 - k)/71)^n, {k, 1, 71}]]}, {n, 1, 500}];
sol = Select[data, #[[2]] >= 1/2 &][[1]]
(* {329,0.5036576494672924} *)
So the n you're looking for is 329.
A better way is to use the While function:
n = 1;
While[Sum[(-1)^(k - 1)*Binomial[71, k]*(1 - (71 - k)/71)^n, {k, 1, 71}] < 1/2, n = n + 1]
n
(* 329 *)
N[Sum[(-1)^(k - 1)*Binomial[71, k]*(1 - (71 - k)/71)^n, {k, 1, 71}]]
(* 0.5036576494672924 *)
