Each round consists of flipping all of the coins showing heads.

If fewer than half of the flipped coins come up heads the player loses.

Rounds continue until the player either loses or has one heads remaining.

The player wins by getting to one heads without losing.

What is the probability of winning this game?

Examples: 6→4→1 would be a loss. (1 is less than half of 4.) 6→5→3→2→2→2→1 would be a win.

   5   point 10
  10    print 1-fnPLose(6),(1-fnPLose(6))/1
  20    end
  30
  40    fnPLose(N)
  50       local Pl,H,Ph,Totph
  60     Pl=0:if N=1 then goto 150
  65     Totph=0
  70     for H=0 to N-1
  80       Ph=combi(N,H)//2^N
  85       Totph=Totph+Ph
  90       if H>=N//2 then
 100        :Pl=Pl+Ph*fnPLose(H)
 110       :else
 120        :Pl=Pl+Ph
 130       :endif
 140     next H
 145     Pl=Pl//Totph
 150    return(Pl)
 
Note that the situation of all the tossed coins coming up heads had to be ignored to avoid an endless regression.  This is legitimate as the transition to a lower value is essentially inevitable, and the normalization of the probabilities takes place in line 145 where the probability of losing from a given number of heads is increased by the factor of the reciprocal of the total probabilities accounted for by situations of a decrease in the number of heads.
 
The program finds

 52//279         0.186379928315412186379928315412186379928315412185
 
meaning the probability is 52/279, or 0.186379928315412..., with the ellipsis starting the endless repetition with the 1863.

Simulation verification:

FOR tr = 1 TO 1000000
  n = 6
  loss = 0: win = 0
  DO
    hds = 0
    FOR c = 1 TO n
      h = INT(RND(1) * 2)
      hds = hds + h
    NEXT
    IF hds < n / 2 THEN loss = 1 ELSE IF hds = 1 THEN win = 1
    n = hds
  LOOP UNTIL loss = 1 OR win = 1
  losses = losses + loss
  wins = wins + win
  PRINT wins, losses, tr
NEXT tr

does a million trials, resulting in 186865 wins.