Flip a coin 6 times. Probability with past results and probability without past results are different?

Question

I am closed in a room and decide to flip a fair coin 5 times (1/2 probability of H and T) and get 5 heads HHHHH so I call a friend to join me inside and he does not know I had flipped the coin 5 times.

If I flip the coin now the 6th time, for my friend (for him it is the first flip) the chance of having head H would be 1/2, for me is it the same 1/2? But actually the probability of having 6 heads in a row on 6 flips is not 1/2.

If we play and I bet that the next flip is tail T, I will win more times then him.

I actually simulated this on 4,459 series of 6 flips 431 times I win the game (6th flip has tail T) and 75 times my friend wins (6th flip has H).

The interesting thing is that for my friend, that does not know about the past results, the probability is the same as it is on one coin flip game so 1/2.

Which is the real probability to have H heads at the 6th flip of 6 coins after 5 heads HHHHH?

Answer 1

For a fair coin, after you have $5$ or $1000$ H the probability to have another H is exactly $\frac12$.

Otherwise when you start the experiment the probability to have 6 H in a row is $\left(\frac12\right)^6$.

Answer 2

4,459 series of 6 flips 431

There are only $2^6=64$ distinct series of 6 flips.

The answer to your question is subtle: On the one hand, getting HHHHH should count as Bayesian evidence against the coin being unbiased. But if it actually is a fair coin, then no, you will not win more often than your friend.

Answer 3

A coin never remembers (in practice there is some bias toward the side facing up when you start the flip because no one can ever flip a coin entirely straight, but that's minor and irrelevant to this discussion), so no matter what it has flipped before and whether your friend knows about them, the odds are even for you and your friend.

That being said, you have another, more subtle misconception here. You seem to think that for any given event, there is a fixed probability, and therefore you think it's strange when two people get different probabilities for the coin coming up heads. That's not true. Probabilities depend a lot on what we know. For instance, if I shuffled a deck of cards, then looked at the bottom card (but didn't show it to you), then the probability of the top card being an ace of spades is different for me than it is for you. For me it's either $0$ or $\frac{1}{51}$, while for you it's $\frac{1}{52}$.