One guy tosses a coin infinite times. The coin is fair, so chances of seeing a head (H) or a tail (T) are equal. What is the probability that he observes a TT before a HT?
Wouldn't the probability of observing TT before HT be 50%?
Since the tosses are iid and the coin is fair, prob(TT) = .5*.5=.25 and prob(HT) = .5*.5=.25
You can think of the infinite sequence of coin tosses as an infinite sequence of pairs of coin tosses, where each of the 4 possible pairs (TT, TH, HT, HH) occurs with equal probability. So then it comes down to what's the probability of observing one of these 4 pairs before another, and since they are equally likely, it should be 50%.