Final answer:
The probability that Nicole wins eight points in a row is f^8, and the probability that Nicole loses nine points in a row is f^9. For Blake to win the game after four points, when his probability of winning a point is 0.2, the probability is 0.2^4 or 0.002 when rounded to the nearest thousandth.
Step-by-step explanation:
If Nicole is playing in a tennis match and the probability that she wins a given point is P(Nicole wins a point) = f, the probability that Nicole wins eight points in a row would be f^8 where f is the probability of her winning a single point. Conversely, if the probability that Nicole loses a given point is also f, then the probability that Nicole loses nine points in a row would be f^9.
When considering the match between Blake and Nicole, if the probability that Blake wins a point is P(Blake wins a point) = 0.2, and the probability that Nicole wins a point is P(Nicole wins a point) = 0.8, the probability that after four points Blake will have won all of them (and therefore the game) would be 0.2^4 or 0.0016 which rounds to 0.002 when rounded to the nearest thousandth.