Problem 4

Alex and Blair are watching their friend Casey practice basketball shots. To pass the time, they come up with a betting game: if Casey scores 3 times in a row, Alex wins. If Casey scores, then misses the next shot, then scores again, Blair wins. They keep watching until one of these two possibilities happen. If each shot of Casey is successful with the same non-zero probability $p$, for which $p$ is the bet fair? Give your answer as a decimal rounded to three decimal places.