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

If an even number of coins is flipped and an even number of heads comes up the player loses.

If an odd number of coins is flipped and an odd number of heads comes up the player loses.

Rounds continue until the player either loses or reaches zero heads.

The player wins by getting to zero heads without losing.

What is the probability of winning this game?

Examples:
6→3→1 would be a loss.
6→3→0 would be a win.
6→0 would be a loss (even→even rule is applied first.)

    5   point 10
    9   for Nbr=6 to 1 step -1
   10    print Nbr,1-fnPLose(Nbr),(1-fnPLose(Nbr))/1
   11   next Nbr
   20    end
   30
   40    fnPLose(N)
   50       local Pl,H,Ph,Totph
   60     Pl=0:if N=0 then goto 150
   70     for H=0 to N
   80       Ph=combi(N,H)//2^N
   90       if H @ 2<>N @ 2 then
  100        :Pl=Pl+Ph*fnPLose(H)
  110       :else
  120        :Pl=Pl+Ph
  130       :endif
  140     next H
  150    return(Pl)

The above program, working recursively, again fills in Steve Herman's list of probabilites when starting with different numbers of tossed coins. In descending order, they are:

6       16793/131072   0.12812042236328125
5       563/4096       0.137451171875
4       23/128         0.1796875
3       7/32           0.21875
2       1/4            0.25
1       1/2            0.5

and, since the puzzle starts with 6 coins, the solution is 16793/131072 = 0.12812042236328125. All the decimal fractions are exact and terminating as all the denominators are powers of 2.

Simulation verification:

A million trials, via

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 MOD 2 = n MOD 2 THEN loss = 1 ELSE IF hds = 0 THEN win = 1
    n = hds
  LOOP UNTIL loss = 1 OR win = 1
  losses = losses + loss
  wins = wins + win
  PRINT wins, losses, tr
NEXT tr

result in 127731 wins.