Consider a game with N groups of people, where each group has 8 people. Each round, every person in every group chooses H or T. A leader flips a fair coin. If a person guessed incorrectly, they are out of the game. Repeat rounds until there is one single winner standing. Their group wins.
- What is the probability that any one group will win?
- Is it better for a group to evenly split their guesses (for example, 4 people guess H and 4 people guess T), or to go all-in on one guess for a round?
- What are some strategies for a group to increase their probability that they will win?
Note: this is a real world game I've played, not a problem for school or out of a textbook, I'm a few years out of school now and pretty rusty at probabilities.
Edit: I'm also interested in a general case where up to 8 players could be on a team.