Problem: Lebron and Stephen are playing a game in which they each take turns shooting a basketball from the 3-point line. Anytime someone misses their shot, the other player then attempts his. This will continue, alternating between the two players, until a shot is made. The person who first makes his shot is declared the winner.
Assume that Lebron has a 60% chance of making any shot, independent of any other shot, and that Stephen has a 70% chance of making any shot, independent of any other shot.
If Lebron attempts the first shot, what is the probability that he will win the game? Round your answer to the nearest ten-thousandths place.
(Hint: Consider the different ways in which Lebron wins, and find the sum of the probabilities using the infinite geometric series formula.)