It seems to me as if you have misunderstood this game.

On any one tern there are only eight outcomes.

(H,H,H),(H,H,T),(H,T,H),(H,T,T),(T,H,H),(T,H,T),(T ,T,H),(T,T,T)

In all but two of those there will be anodd man out

(H,H,H),(H,H,T),(H,T,H),(H,T,T),(T,H,H),(T,H,T),(T,T,H),(T,T,T)

Thus the probability ofodd man outon any turn is .

In order that the firstodd man outhappens on the second turn is .

In order that the firstodd man outhappens on the fourth turn is .

Can you finish?