There're two unbalanced coins loaded in a way such that, when they are tossed, one comes up a head with probability 1/3 and the other 3/7 . You are offered the following game:
The entrance fee is $1 and these two coins are tossed for 441 times with the number of times of observing two heads or two tails counted. If this number is more than 252 or less than 210, you win $10.
a. What is the distribution that, Y = the number times of observing two heads or two tails, is following?
b. Write down the exact formula for your probability of winning and losing.
c. Make your decision on wether to play or not and justify your answer by approximating the expected winning per game.
d. According to your approximation in c, what is the "break-even" entrance fee?