Guessing a deck of three aces and three jacks

A face-down pile contains exactly $3$ aces and $3$ jacks in random order, and Jamie knows this. Cards are turned over one at a time; before each card is revealed Jamie guesses whether it is an ace or a jack, winning $\text{\char36}1$ for each correct guess. Playing all $6$ cards optimally, how much does Jamie expect to win?

$\text{\char36}4.10$. Always guess the type with more cards remaining (either on a tie). By backward induction, or by summing the turn-by-turn success probabilities $\tfrac12,\tfrac35,\tfrac35,\tfrac{7}{10},\tfrac{7}{10},1$, the expected total is $\tfrac{41}{10}$.