$10$ poker, $5$ of them are red, the rest of them are black. Randomly shuffled.
The gamer is asked to guess the color of each poker sequentially. After each guess, he will be told the correct color, i.e. the gamer will know the color of the $i$-th poker before he guesses the color of the $i+1$-th. If all of his guesses are right, he will win a prize of $\$1000$.
What is the optimal strategy? What is the fair price of this game?
Any thoughts? Thanks!
It makes no difference to tell the gamer whether each color is correct as the game progresses, because the only payout is for an all-correct answer.
There are $\binom {10}{5}=252$ possible color mix orders from the specified $5$ red and $5$ black set given, so assuming that the gamer is sufficiently rational to pick a mix of $5$ red and $5$ black, they will have a $1$-in-$252$ chance of the payout.
The expected payout is therefore $\frac{\$1000}{252} \approx \$3.97$. As the setter, I would say that a $\$4$ game fee would probably give a "house advantage" that was slightly too low. Knowing human psychology I'm sure you could get away with a $\$5$ game fee because the $\$1000$ prize is big enough to quell serious calculation. :-)