The expected value of the game is between $0 and $10, depending on how many coins you re-toss.
The expected value of a game is the average of the values of each possible outcome. Since there are two possible outcomes in this game (either you win or you lose), we can calculate the expected value by averaging the value of each outcome.
The value of winning the game is $10. The probability of winning the game is the probability of getting all three coins the same way up on the first toss, plus the probability of getting all three coins the same way up after the re-toss.
The probability of getting all three coins the same way up on the first toss is 1/8, since there are 8 possible outcomes (HHH, HHT, HTH, THH, TTH, THT, HTT, TTT), and only one of them results in all three coins being the same way up.
The probability of getting all three coins the same way up after the re-toss is more complicated to calculate, but it is still less than 1/2. This is because there are more ways to get three different outcomes than there are to get all three coins the same way up. For example, there are 3 ways to get HHH, HHT, HTH, THH, TTH, THT, HTT, TTT, but there are only 3 ways to get all three coins the same way up.
Therefore, the expected value of the game is less than $10, since the probability of winning the game is less than 1/2. However, the expected value is still greater than $0, since there is a chance that you will win.
To calculate the exact expected value, we need to know the exact probability of getting all three coins the same way up after the re-toss. This probability depends on how many coins you re-toss. If you re-toss all three coins, then the probability is 1/8. If you re-toss only two coins, then the probability is 3/8. If you re-toss only one coin, then the probability is 5/8.
Therefore, the expected value of the game is between $0 and $10, depending on how many coins you re-toss.