A game is played starting with 6 fair coins laid out with heads face up.

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.)

I think you need to work backwards.

If there is one head (and you haven't already lost),

then on the next flip the probabilities are as follows:

1/2 -- 0 heads -- you win

1/2 -- 1 head -- you lose

Probability of winning = 1/2

If there are 2 heads (and you haven't already lost),

then on the next flip the probabilities are as follows:

1/4 -- 0 heads -- you lose

2/4 -- 1 head -- you win with probability 1/2

1/4 -- all heads -- you lose

So, if there are 2 heads (and you haven't already lost), then your probability of winning is 1/4 (i.e, 1/2 * 0 + 1/2 *1/2).

If there are 3 heads (and you haven't already lost), then on the next flip the probabilities are:

1/8 -- 0 heads -- you win

3/8 -- 1 head -- you lose

3/8 -- 2 heads -- you win with probability 1/4

1/8 -- 3 heads -- you lose

So, if there are 3 heads (and you haven't already lost), then your probability of winning is 7/32 (i.e, 1/2 * 0 + 1/8*1 + 3/8 *1/4).

If there are 4 heads (and you haven't already lost), then on the next flip the relative probabilities are:

1/16 -- 0 heads -- you lose

4/16 -- 1 head -- you win with probability 1/2

6/16 -- 2 heads -- you lose

4/16 -- 3 heads -- your win with probability 7/32

1/16 -- 4 heads -- you lose

So, if there are 4 heads (and you haven't already lost), then your probability of winning is 23/128 (i.e, 4/16*1/2 + 4/16 *7/32 = 1/4*(23/32)), unless I just made a mistake.

Gotta go, but working backwards to 5 and then 6 will yield the required probability. I can already tell, though, that I don't like my chances of winning.