A fair coin happens if the coin has probability 1⁄2 of success on each trial. We know nothing is perfect, including the weight distribution of coin. Hence, the probability of end up on either head or tail is not exactly 1⁄2. But, I think there is a better way to toss a coin:
1. Pick head or tail 2. Do the toss twice 3. If both tosses differ, only use the first toss. But if both tosses match, try again.
Suppose that person A picks head, person B picks tail. The coin was tossed twice, the first toss was head, the second was head (HH). In this case, no one wins. But then if the first toss was tail, the second was head (TH). B wins. Conversely, if the first toss was head and the second was head (HT), then A wins.
Tossing the coin this way is to make sure either head or tail has the same exact probability. We know that first toss and second toss are independent, hence
P(A and B) = P(A)P(B). Let’s say we have an unfair coin with probability of getting head is 0.6 and tail is 0.4. Then,
P(H)P(H) = 0.6 * 0.6 = 0.36 P(H)P(T) = 0.6 * 0.4 = 0.24 P(T)P(H) = 0.4 * 0.6 = 0.24 P(T)P(T) = 0.4 * 0.4 = 0.16
In the example above we can see HT and TH have the same probability of happening. Tossing the coin twice is to make unfair coins fair, or I’d say it is the more proper way to toss a coin.