I have this problem which i think I have solved but i am not quite sure if it is correct
Consider an urn containing one white and one black ball. At each step a ball is selected uniformly at random from the urn. It is then returned to the urn along with an additional ball of the same colour.
Let $X_n$ be the proportion of white balls in the urn after the n-th step.
(a) Show that {$X_n, n = 1, 2,... $} is a martingale.
(b) Show that the probability that $X_n$ ever exceeds 0.55 is no greater than $10/11$
I have solved a) already and I found the correct answer to b) but I am not sure if i used the correct reasoning, mainly because $X_n$ is a martingale and the theorem i used is for sub-martingales.
So i used the fact that if {$X_n, n = 1, 2,... $} is a non-negative sub-martingale then
$P(max(X_1,...,X_n) > a) \leq E(X_n)/a$
And then $P(max(X_1,...,X_n) > 0.55) \leq 0.5/0.55 = 10/11$
Where I found $E(X_n) = E(X_1) =0.5 $ since obviously at step 1 there are 2 balls of which one is white [this is correct yes?]
Now I am just unsure if this method can in fact be used for normal martingales since my texbook does not state it. Should i have used a completely diffrent method and it is only by chance i got the correct answer?