Answer:
The casino should charge for this game at least $1 to break even.
Explanation:
We can define the prize function as

where M is the prize money and n is the number of tails in continous flips.
The probability of n consecutive tails can be calculated as
. The probaility of getting a head after the n consecutive tails is
, so the probability of having n consecutive tails and a head is

Then we can calculate the expected value of M as

The expected money prize for this game is $1, so the casino should charge to play at least $1 to break even.