a coin is tossed until ahead appears twice in arow.what is the sample space for this experiment?if the coin is fair.what is the probability that it will be tossed exactly four times?

The sample space denotes all the possible outcomes.

The event where it is tossed 4 times is as follows:
(I'll use H for head and T for tail)

HTHH
TTHH

Those are the only way. Now, can you work up the probability? (Hint, probability of having a head is 0.5, that of having a tail is 0.5 too)