Final answer:
The answer involves calculating the probability of flipping exactly one head with three fair coins, which is 3/8. It also differentiates between fair and biased coins and explains how to determine the expected payoff when using biased coins.
Step-by-step explanation:
The question asks for the probability of flipping 3 fair coins and getting exactly 1 head. To solve this, we need to understand the concept of theoretical probability and combinatorial analysis. When flipping one fair coin, the theoretical probability of getting heads is 0.5. Flipping three coins increases the number of possible outcomes.
The total number of outcomes when flipping three coins is 2^3 = 8, because each coin has two possible outcomes (heads or tails) and there are three coins. The desired outcomes (exactly one head) can occur in three ways: H T T, T H T, or T T H. Therefore, the probability of getting exactly one head is the number of desired outcomes divided by the total number of outcomes, which would be 3/8.
Understanding the outcome of biased coin flips is a different task. In the case of a biased coin where the probability of heads is 3 times the probability of tails, the expected payoff per game can be determined. This can be found by multiplying the probability of each outcome by the corresponding payoff and then summing these values.