You are playing with some friends this summer. The game is simple: a fair coin is picked and tossed. Before each toss, you and your friends bet on either heads or tails coming up. Say you begin playing at 8 in the afternoon, and time flies till 12 while playing and chatting.

However something interesting happened in those 4 hours. You tossed the coin around 200 times, and out of those 200 times, heads came up 180 times, surprisingly, but possible. Once again, one friend is an engineer and he certified the coin as fair. In the $201$st round, just before going home, everyone decides to do an all-in for the last toss. You need to decide where to put all your money, or how to win as much as possible. Which side should you trust?

---

In the analysis I didn't the result did convince me, but in the end the coin tosses should obey a $X \sim Bi(n=201 , p=\frac{1}{2})$, and we have to find $P(X=181 \mid X \ge 180)$ so the probability should be 

$$
P(Y\_{201}=heads)=
\frac{P(X=181)}{P(X \ge 180)}=
\frac{\binom{201}{181}\frac{1}{2}^{201}}{1-F(180)}=
.8908
$$

But shouldn't the last toss be independent of the of history of the game? If another friend arrived just for the last toss, he sees the probability as one half. I'm not sure of how to interpret this situation! Maybe I did something wrong with the binomial or missed some concept.