y=1900; m=1; d=1;

ct=0; yList=[]; considered=0;

dt=datetime(y,m,d);

while y<1952

  dt2=datetime(y+50,m,d);

  considered=considered+1;

  if month(dt2)==m % leap->non-leap gives March vs Feb

    if weekday(dt)==weekday(dt2)

      ct=ct+1;

      if ~ismember(year(dt),yList)

        yList(end+1)=year(dt);

      end

    end

  end

  jd=juliandate(dt)+1;

  dt=datetime(jd,'ConvertFrom','juliandate');

 

  y=year(dt); 

  m=month(dt);

  d=day(dt);

end

ct

considered

ct/considered

yList

finds

ct =

        9431

considered =

       18992

ans =

          0.49657750631845  == the desired probability

          

Fifty years can be 18262 or 18263 days depending on the number of leap year days intervening--either 12 or 13. 18263 is divisible by 7, and so all the dates in a 1-year span of March through February will or will not be weekday matches. Leap year days will definitely not be matches, as 50 years later will not be another leap year. This 4-year cycle is certainly a basis for the major part of an analytic solution, but with consideration for the fact that 1900 was not a leap year although it was divisible by 4, so that the first two months of that year are not in sync with the cycle.