#x86_64 machine language for Linux, <s>15 19 17</s> 16 bytes

    L1:
    48 0f c7 f0             rdrand %rax
    f3 48 0f b8 c0          popcnt %rax,%rax
    3c 1a                   cmp    $0x1a,%al
    7d f3                   jge    L1
    04 41                   add    $0x41,%al
    c3                      retq
This requires support for the POPCNT and RDRAND instructions.

A uniform distributed random number is generated, the number of 1's in that number is counted, if that number is less than 26, a letter is returned. One will need to let the code run a long time before one sees a letter A.

To test, try something like

    #include<stdio.h>
    #define TEST "\x48\xf\xc7\xf0\xf3\x48\xf\xb8\xc0\x3c\x1a\x7d\xf3\4\x41\xc3"
    int main(){
      int hist[26]={0};
      for(int i=0;i<10000000;i++){
        hist[ ((int(*)())TEST)() - 'A' ]++;
      }
      for(int i=0;i<26;i++){
        printf("%c %d\n", 'A'+i, hist[i] );
      }
    }
Sample output

    A 0
    B 0
    C 0
    D 0
    E 0
    F 0
    G 0
    H 0
    I 0
    J 0
    K 0
    L 8
    M 32
    N 137
    O 511
    P 1639
    Q 5188
    R 14475
    S 37539
    T 91670
    U 205638
    V 431381
    W 842259
    X 1536776
    Y 2626524
    Z 4206223
The analytical expression for the probability of each letter can be derived from the binomial distribution. The letter A is assigned index k=0, B is assigned k=1 and so on.

            /  \
           | 64 |
           | k  |
            \  /
    p(k)=------------
          25
          --- /  \
          \  | 64 |
          /  | i  |
          --- \  /
          i=0

     p(A)~1.0483e-18
     p(B)~6.7093e-17
     p(C)~2.1134e-15
     p(D)~4.3678e-14
     p(E)~6.6608e-13
     p(F)~7.9930e-12
     p(G)~7.8598e-11
     p(H)~6.5124e-10
     p(I)~4.6401e-09
     p(J)~2.8872e-08
     p(K)~1.5879e-07
     p(L)~7.7953e-07
     p(M)~3.4429e-06
     p(N)~1.3772e-05
     p(O)~5.0169e-05
     p(P)~1.6723e-04
     p(Q)~5.1214e-04
     p(R)~1.4460e-03
     p(S)~3.7758e-03
     p(T)~9.1413e-03
     p(U)~2.0568e-02
     p(V)~4.3095e-02
     p(W)~8.4231e-02
     p(X)~1.5381e-01
     p(Y)~2.6276e-01
     p(Z)~4.2042e-01