There are n women and n men who are all together and go one being formed completely randomly in a row to enter an audience. Before of the entrance, and once they are formed, each person turns to see the person front and back (if there are both or only in the case of those who are left at the extremes of the row). If your partner is from a different gender, you are greeted with a kiss and if they are of the same gender they are greet each other with a hug.
a) Calculate the expected number (depending on n) of greetings of kisses there will be.
b) Calculate the expected number (depending on n) of greetings with a hug that will be
c) After doing some relaxation they re-form way completely random but now what they want is to put combos of cookies and coffee to auditory so that each one of three consecutive of them there is a boy and two girls (in any order) or a girl and two children (also in any order). they agree and put a combo of coffee and cookies to the audience. What is the expected amount of combos that enter the auditorium?
Sorry for the bad traduction.
For parts a) and b), consider that every person, except the rightmost, greets the person on his right, and there are $2n-1$ greetings in all. Now for any particular person, the probability that the person on his right is of the same sex is $\frac{n-1}{2n-1}$ and the probability that the person is of the opposite sex is $\frac{n}{2n-1},$ so the given person's expectation of a hug is $\frac{n-1}{2n-1}$ and his expectation of a kiss is $\frac{n}{2n-1}.$ By linearity of expectation, the expected number of hugs is $n-1,$ and the expected number of kisses is $n.$
Now try the same idea on part c).