Suppose Serena and Rafael are playing tennis and the score is deuce, meaning that the next player to go ahead by two points wins the game. Suppose that Serena is serving, that Serena wins points when serving with probability p, and that the outcomes of points are independent. What is the probability that Serena wins the game? Solution when serving serena wins with probability = p so for here to win probability = win two points back to back + lost first point and then win 2 points back to back (if she loses first 2 points she looses) so probabitiy wins two points back to back is = p*p probability she loses first point and wins two points back to back is (1-p)*p*p so total probability is = (p*p)+((1-p)*p*p) = (p^2)* (1+(1-p)) =(p^2)* (2-p)).

1. Suppose Serena and Rafael are playing tennis and the score is deuce, meaning that the next
player to go ahead by two points wins the game. Suppose that Serena is serving, that Serena wins
points when serving with probability p, and that the outcomes of points are independent. What is
the probability that Serena wins the game?
Solution
when serving serena wins with probability = p so for here to win probability = win
two points back to back + lost first point and then win 2 points back to back (if she loses first 2
points she looses) so probabitiy wins two points back to back is = p*p probability she loses first
point and wins two points back to back is (1-p)*p*p so total probability is = (p*p)+((1-p)*p*p)
= (p^2)* (1+(1-p)) =(p^2)* (2-p))