James bought \(n\) pairs of identical socks. For \(n\) days James did not have any problems: every morning he took a new pair of socks out of the closet and wore it all day. After \(n\) days, James’ father washed all of the socks in the washing machine and put them into pairs in any way possible as, we repeat, all of the socks are the same. Let’s call a pair of socks successful, if both socks in this pair were worn by James on the same day.
a) Find the probability that all of the resulting pairs are successful.
b) Prove that the expectation of the number of successful pairs is greater than 0.5.