Random questions from my math class

-readdresses some of the already tackled problems-

 (9)

Conditional probability: P(A given B) = P(A and B)/P(B)
Since there are 6 sides, 3 of which are nude, and the side that is visible is chosen at random, there is a 1/2 chance of the visible being nude [P(B) = 1/2].
The probability that both sides are nudes (and, following from that, the visible is also nude) is 1/3.
Therefore, P(other_nude given visible_nude) = (1/3)/(1/2) = 2/3.

 (11)

The problem is that the distribution of Y is assumed to be a uniformly random positive number. This also means that Y's average value is infinite. Infinity*5/4 is still infinite, so that conclusion would indeed hold. But, all this is illogical anyway because no one has an infinite amount of money to give.

 (13)

Say you start off with the billiard ball numbered n. There are n-1 balls less than n, and 15-n balls greater than n.
Due to the restrictions on the problem, once you've selected your first billiard ball, there are only two legal "moves" at any point: knock off the ball numbered closest to but less than n (let's abbreviate this action L), or knock off the ball numbered closest to but greater than n (abbreviated R).

Therefore the number of moves starting with n, is simply the number of ways to permute n-1 instances of L and 15-n instances of R, or 14 choose n-1.

Summing up the 15 possibilities for the starting value of n, this is sum(i = 0 to 14) 14 choose i, which equals 2^14 = 16384.

 (14)

Every 400 years, there are 100-4+1 = 97 leap years, and 303 normal years. This is 97*366+303*365 = 146097 days. Since this divides into 7 evenly (146097/7 = 20871), the probability that any given day in this 400-year span is 1/7. This means that this 400-year span can (excuse the non-technical language) be copypasta'd to the rest of history without biasing the distribution of weekdays.
However, in 400 years, there are 400 Christmases. 400 cannot be evenly divided into 7, therefore the probability that any given Christmas in the 400-year span (and therefore the rest of history) falls on a Wednesday cannot be 1/7.

 (16B)

Her base camp is 40 miles away from the south pole in any direction. Going "east" at the south pole amounts to spinning in circles.

 15

31 in octal (base 8) is 25 in decimal (base 10). 3*8+1 = 24+1 = 25.

 17

Not having a Friday the 13th in a month is not an independent event. I would assume from the given information that they occur approximately once every 7 months.

Would calculate, but meh.

 19

A B
C D
E F
G G