ClearAll[PlayRandom, PlayOptimal]
PlayRandom[n_] :=
 Module[{pardoned = 0, sampler, indrawer, found, reveal},
  sampler = indrawer = Range[100];
  Do[
   indrawer //= RandomSample;
   found = 0;
   Do[
    reveal = RandomSample[sampler, 50];
    If[MemberQ[indrawer[[reveal]], p],
     found++;
     ]
    ,
    {p, 100}
    ];
   If[found == 100, pardoned++];
   ,
   {n}
   ];
  N[pardoned/n]
  ]
PlayOptimal[n_] :=
 Module[{pardoned = 0, indrawer, reveal, found, card},
  indrawer = Range[100];
  Do[
   indrawer //= RandomSample;
   Do[
    reveal = p;
    found = False;
    Do[
     card = indrawer[[reveal]];
     If[card == p,
      found = True;
      Break[];
      ];
     reveal = card;
     ,
     {g, 50}
     ];
    If[! found, Break[]];
    ,
    {p, 100}
    ];
   If[found, pardoned++];
   ,
   {n}
   ];
  N[pardoned/n]
  ];
PlayRandom[1000]
PlayOptimal[10000]